Tables of the Appell Hypergeometric Functions $ F_2$

Tables of the Appell Hypergeometric Functions F2 Jonathan Murley and Nasser Saad∗

arXiv:0809.5203v3 [math-ph] 28 Oct 2008

Department of Mathematics and Statistics, University of Prince Edward Island Charlottetown, Prince Edward Island C1A 4P3, Canada Abstract: The generalized hypergeometric function q Fp is a power series in which the ratio of successive terms is a rational function of the summation index. The Gaussian hypergeometric functions 2 F1 and 3 F2 are most common special cases of the generalized hypergeometric function q Fp . The Appell hypergeometric functions Fq , q = 1, 2, 3, 4 are product of two hypergeometric functions 2 F1 that appear in many areas of mathematical physics. Here, we are interested in the Appell hypergeometric function F2 which is known to have a double integral representation. As demonstrated by Opps, Saad, and Srivastava (J. Math. Anal. Appl. 302 (2005) 180-195), the double integral representation of F2 can be reduced to a single integral that can be easily evaluated for certain values of the parameters in terms of 2 F1 and 3 F2 . Using many of the reduction formulas of 2 F1 and 3 F2 and the representation of F2 in terms of a single integral, we have begun to tabulate new reduction formulas for F2 .

PACS: Primary 33C65, 33C05, 33D15, 33D60; Secondary 33B15, 33C20, 33D90. keywords: Multiple hypergeometric functions, Appell series, Generalized hypergeometric function, Gauss hypergeometric function, Clausen hypergeometric function, Reduction and transformation formulas, Radiation field, Hubbell integral. I.

INTRODUCTION

Appell hypergeometric functions FD , D = 1, 2, 3, 4 play an important role in mathematical physics ([1]-[3], [5], [6][7], [31], [32]). In particular, the Appell hypergeometric series F2 arises frequently in various physical and chemical applications ([4], [8] to [26], [28] to [35] ). The exact solution of number of problems in quantum mechanics has been given [24] in terms of Appell’s function F2 . It is defined by [[31], p. 211, Equation (8.1.4)]: ∞ X ∞ X (σ)m+n (α1 )m (α2 )n xm y n F2 (σ, α1 , α2 ; β1 , β2 ; x, y) = , (β1 )m (β2 )n m! n! m=0 n=0

(1)

− for |x| + |y| < 1; βj ∈ C\Z− 0 ; Z0 := {0, −1, −2, . . .}, and (λ)k denotes the Pochhammer symbol defined, in terms of Gamma functions, by  Γ(λ + k) 1 if (k = 0; λ ∈ C\{0}) = (λ)k := λ(λ + 1)(λ + 2) . . . (λ + k − 1) if (k ∈ N; λ ∈ C) Γ(λ)

where N being the set of positive integers. Further, it also has the following double integral representation [[31], p. 214, Equation (8.2.3)]: Z 1Z 1 Γ(β1 )Γ(β2 ) F2 (σ, α1 , α2 ; β1 , β2 ; x, y) = uα1 −1 τ α2 −1 Γ(α1 )Γ(α2 )Γ(β1 − α1 )Γ(β2 − α2 ) 0 0 ×(1 − u)β1 −α1 −1 (1 − τ )β2 −α2 −1 (1 − xu − yτ )−σ du dτ

(2)

where R (βj ) > R (αj ) > 0, j = 1, 2 and |x|+|y| < 1. Recently, Opps et al. [26] used the Euler’s integral representation of the Gauss hypergeometric function [[31], p. 20, Equation (1.6.6)]   Z 1 a, b Γ(c) τ b−1 (1 − τ )c−b−1 (1 − zτ )−a dτ (3) z = 2 F1 Γ(b)Γ(c − b) 0 c to reduce the double integral representation (2) into a single integral in term of 2 F1 ,   Z 1 α1 −1 σ, α2 y Γ(β1 ) u (1 − u)β1 −α1 −1 du F F2 (σ; α1 , α2 ; β1 , β2 ; x, y) = 2 1 β2 1 − xu Γ(α1 )Γ(β1 − α1 ) 0 (1 − xu)σ

∗ Electronic

address: [email protected]; Electronic address: [email protected]

(4)

2 for R (β1 ) > R (α1 ) > 0, and |x| + |y| < 1. Using some properties of 2 F1 , they prove the following theorem [[26], Theorem 1]: Theorem 1. For |x| + |y| < 1, the Appell hypergeometric function F2 is given by     1 1 a, α1 a, α1 x F2 (a + 1; α1 , 1; β1 , 2; x, y) = − x + F 2 F1 2 1 β1 β1 1 − y ay ay(1 − y)a

(5)

where a 6= 0; α1 ∈ C; β1 ∈ C\Z− 0 and

     α1 + 1, 1, 1 α1 x α1 + 1, 1, 1 x ln(1 − y) − x F F x − 3 2 3 2 β1 + 1, 2 β1 y 1 − y y β1 + 1, 2 1 − y

F2 (1; α1 , 1; β1 , 2; x, y) =

(6)

C\Z− 0.

where α1 ∈ C; β1 ∈ The present work is devoted to compute F2 (σ; α1 , 1; β1 , 2; x, y) explicitly for different values σ, α1 , β1 of the function parameters using (5) and (6). This is mostly done using many of the reduction formulas of 2 F1 and 3 F2 listed in [27] and other sources of special functions ([5], [6]-[7], [31], [32]), we begun here to tabulate reduction and transformation formulas for F2 . First, in Tables I and II, we give the corrections to some formulas misprinted in the classical monograph by Prudnikov et al. [27]. TABLE I: Correction to some formulas for 2 F1 reported in the classical work of Prudnikov et al [27]. σ

α

β

5 2 4 5

4

1

1

14 5

5 6

1

17 5

1

7 2

9 2

1

b 1−n 2

b−m

− n2

2 F1 (σ, α; β; z) 11 1 (16 + 72z + 18z 2 − z 3 )(1 − z)− 2 16 √ ˆ −1 √ ( 5−1)x+x2 5 5 9 9 √ − 25x − 5 ln 1−2 9 (1 − x ) ln(1 − x ) − 5 ln(1 − x) − 5x5 1+2−1 ( 5+1)x+x2 √ 1 √ 1 √ 1 √ 1 (10+2 5) 2 x (10−2 5) 2 x ˜ 2(10 + 2 5) 2 arctan 4−(√5−1)x − 2(10 − 2 5) 2 arctan 4+(√5+1)x

1

[x = z 5 ]

1 √ ˜ 6 1 1−x+x2 55 1−x 32 x 3 arctan 1−x + 36x 11 (1 − x )(ln 1+x + 2 ln 1+x+x2 + 2 √ ˆ ˜ 15 tanh−1 z 2 7 √ - 15z 3 15 + 15z + 3z − z Pm−1 (−m)k −k−1 b−m−1 m! (z − 1)−m−1 − (1−b) k=0 (2−b)k (1 − z) b−1 m √ 2−n (1 + 1 − z)n , (n 6= 1, 2)

11 6x6

1−n

TABLE II: Correction to some formulas for 3 F2 reported in the classical work of Prudnikov et al [27]. a1 1 4

a2 1

II.

a3

b1

b2

3 F2 (a1 , a2 , a3 ; b1 , b2 ; z)

1

5 4

2

1 3z

ˆ

3` ln(1 − z) + z 4 ln

1

1+z 4 1 1−z 4

1

+ 2 arctan z 4

´˜

SPECIAL VALUES OF THE APPELL HYPERGEOMETRIC FUNCTIONS F2

In the next, we tabulate the explicit computations of F2 (σ; α1 , 1; β1 , 2; x, y) for different values of the function parameters σ, α1 , and β1 . Since the role of α1 , β1 , x and α2 , β2 , y in F2 (σ; α1 , α2 ; β1 , β2 ; x, y) are interchanged, similar tables can be obtain for F2 (σ; 1, α2 ; 2, β2 ; x, y). Even-though the table is given only for α2 = 1 and β2 = 2, the table can be used for a wider variety of cases. This can be notice from the following properties. 1. F2 (σ, α1 , α2 ; β1 , β2 ; x, y) = F2 (σ, α2 , α1 ; β2 , β1 ; y, x). y x 2. F2 (σ, α1 , α2 ; β1 , β2 ; x, y) = (1 − x)−σ F2 (σ, β1 − α1 , α2 ; β1 , β2 ; x−1 , 1−x ) = (1 − x − y)−σ F2 (σ, β1 − α1 , β2 − y x α2 ; β1 , β2 ; x+y−1 , x+y−1 ).

3. F1 (α; β, β ′ , γ; x, y) =

 β ′ x y

F2 (β + β ′ ; α, β ′ ; γ, β + β ′ ; x, 1 − xy ) =


y β x

F2 (β + β ′ ; α, β; γ, β + β ′ ; y, 1 − xy ).

3 Further, the analytic expressions presented in the following tables can be useful in many applications of mathematical physics including the computation of the generalized Hubbell rectangular source integrals, elliptic integrals, and the radiation fields ([4], [8] to [26], [28] to [35]).

4 σ

α1

α2

β1

β2

F2 (σ; α1 , α2 ; β1 , β2 ; x, y)

1 2

- 21

1

- 21

2

2 y [(1

1 2

- 21

1

1 2

2

1 2

1 2

1

- 21

2

1 2

1 2

1

3 2

2

1 2

1 2

1

5 2

2

1 2

1 2

1

7 2

2

1 2

1

1

- 12

2

1 2

1

1

1 2

2

1 2

1

1

3 2

2

1 2

1

1

2

2

1 2

1

1

5 2

2

1 2

1

1

3

2

1 2

1

1

7 2

2

1 2 1 2 1 2

1

1

4

2

3 2 3 2

1

- 21

2

1

1 2

2

1 2

3 2

1

5 2

2

1 2

3 2

1

7 2

2

1 2

2

1

- 12

2

1 2

2

1

1 2

2

1 2

2

1

1

2

1 2

2

1

3 2

2

1 2

2

1

5 2

2

1 2 1 2

2

1

3

2

2

1

7 2

2

1 2 1 2

2

1

4

2

5 2

1

- 21

2

1 2

5 2

1

1 2

2

1 2

5 2

1

3 2

2

1 2 1 2

5 2

1

2

3

1

7 2 - 12

2

1

1

− x) 2 − (1 − x − y) 2 ] q  √ √ √ √ 2 x ) 1 − x − 1 − x − y + x(arcsin x − arcsin 1−y y 1 2 − 12 − (1 − y)(1 − x − y)− 2 ] y [(1 − x) √ x √  √ arcsin 1−y 1 arcsin x−(1−y) √ + 1−x y x

−

√

1−x−y



√ x  √ √ √ (1−4x) arcsin x−(1−y)(1−y−4x) arcsin 1−y √ (1 + 2x) 1 − x − (1 − y + 2x) 1 − x − y − x
 (1−4x+8x2 ) arcsin √x−(1−y) (1−y)2 −4x(1−y)+8x2
√ 5 √ − (3 − 10x − 8x2 ) 1 − x 2 64x y 3 x
√  + 3(1 − y)2 − 10x(1 − y) − 8x2 1 − x − y 3 8xy



3

2 y [(1

− x)−1 − (1 − y) 2 (1 − x − y)−1 ] q   √ √ √ 2 x 1 − y − x(tanh−1 x − tanh−1 1−y ) y 1− √ x  √  √ (1−x) tanh−1 x−(1−x−y) tanh−1 1−y 1 √ 1 − y − 1 − y x  3 3 3 4 2 2 2 3xy 1 − (1 − x) − (1 − y) + (1 − y − x) √ x  √ 2  √ (1−x) tanh−1 x−(1−x−y)2 tanh−1 1−y 3 √ 4xy 1 + x − (1 − y + x) 1 − y − x   5 3 5 5 5 16 2 2 2 15x2 y (1 − x) − (1 − x − y) − 1 + 2 x − (1 − y) (1 − y + 2 x) √ √ x
 (1−x)3 tanh−1 x−(1−x−y)3 tanh−1 1−y 5 √ − 3 + 8x + 3x2 24x2 y 3 x
 √ + 1 − y 3(1 − y)2 + 8x(1 − y) + 3x2  7 7 7
5
3
 4 2 2 2 − 28x 1 − (1 − y) 2 + 35x2 1 − (1 − y) 2 35x3 y 8 1 − (1 − x) − (1 − y) + (1 − y − x) 3 2 − 32 − (1 − y)2 (1 − x − y)− 2 ] y [(1 − x)   1−2x 1−y−2x 2 √ −√ y 1−y−x 1−x x  arcsin √x−(1−y)2 arcsin √ 1−y 3 √ − (1 − 4xy x

 √ √ 2x) 1 − x + (1 − y − 2x) 1 − x − y 
√ √ 5 2 ) 1 − x − 3(1 − y)2 − 4x(1 − y) + 4x2 1 − x − y 16x2 y (3 − 4x + 4x √ √ x
 (1−2x) arcsin x−(1−y)2 (1−y−2x) arcsin 1−y √ −3 x 5

2 y [(1

− x)−2 − (1 − y) 2 (1 − x − y)−2 ]
√ q   √ √ 1−y 2(1−y)−3x 1 2−3x x − − 3 x(tanh−1 x − tanh−1 1−y ) y 1−x 1−x−y   2−3x 1 √ √ − 2−2y−3x y 1−y−x 1−x √ x  √  √ (1−3x) tanh−1 x−(1−y−3x) tanh−1 1−y 1 √ 1 − y) + 2y 3(1 − x x   (1+3x)(1−x) tanh−1 √x−(1−y+3x)(1−y−x) tanh−1 √ 1−y √ 3 √ − 1 + 3x + 1 − y(1 − y − 3x) 8xy x  5
3 3 8 2 − (2 + 3x)(1 − x) 2 + (2 − 2y + 3x)(1 − y − x) 2 15x2 y 2 1 − (1 − y) 
√ 5 2 1 − y 3(1 − y)2 − 2x(1 − y) + 3x2 16x2 y 3 − 2x + 3x − √ √ x
 (1−x)2 (1+x) tanh−1 x−(1−x−y)2 (1−y+x) tanh−1 1−y √ −3 x  5
5 5 7
16 2 - 35x − 7x 1 − (1 − y) 2 − (4 + 3x)(1 − x) 2 + (4 − 4y + 3x)(1 − y − x) 2 3 y 4 1 − (1 − y) 2 y [(1

5

5

− x)− 2 − (1 − y)3 (1 − x − y)− 2 ]  2 2 2 3−12x+8x2 − 3(1−y) −12x(1−y)+8x 3 3 3y (1−x) 2 (1−y−x) 2   3−4x 2 √ √ − 3−3y−4x 3y 1−y−x 1−x  3(1−y)3 −x(1−y)2 −10x2 (1−y)+8x3 5 √ − 24x2 y 1−x−y 2 y [(1

7

− x)−3 − (1 − y) 2 (1 − x − y)−3 ]

2 3−x−10x +8x3 √ 1−x

−3

(1−y)3 arcsin

√

x

√ 1−y x

−arcsin

√

x


5 σ α1 α2 β1 β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)
√ q  √ 2 1−y 8(1−y)2 −25x(1−y)+15x2 −1 √ −1 1 1 1 x − tanh + 8−25x+15x 15 3 1 2 x(tanh x) − 2 2 4y 1−y (1−x−y)2 (1−x)2  2 2 1 1 8−24x+15x2 1 1 2 4y − 8(1−y) −24x(1−y)+15x 3 3 2 3 (1−x) 2 (1−y−x) 2
√ x −1 √
  √1−y 13(1−y)−15x (1−y−5x) tanh−1 x 1−y −(1−5x) tanh 13−15x 1 1 3 √ − + 3 3 1 2 2 2 8y 1−x−y 1−x x  4−5x  1 1 √ √ − 4−4y−5x 1 2 2 2y 2 3 1−y−x 1−x
√ x √ √ −(1+6x−15x2 ) tanh−1 x  (1−y)2 +6x(1−y)−15x2 tanh−1 1 5 3 √ 1−y 3 1 2 1 − y(1 − y − 15x) − 1 + 15x − 2 2 32xy x √
1 7 5 2 2 2 1 2 2 64x2 y 1 − y 3(1 − y) + 4x(1 − y) − 15x − 3 − 4x + 15x 2 3
√ x 2 −1 √
 (1−y)2 +2x(1−y)+5x2 (1−x−y) tanh−1 x 1−y −(1+2x+5x )(1−x) tanh √ −3 x  7
3 1 1 4 2 35x43 y 8 1 − (1 − y) 2 − (8 + 12x + 15x2 )(1 − x) 2 2 3
3 + 8(1 − y)2 + 12x(1 − y) + 15x2 (1 − y − x) 2 7

7

1 2

7 2

1

- 21 2

2 y [(1

1 2

7 2

1

1 2

2

2 5−30x+40x2 −16x3 5 5y (1−x) 2

2

 15−40x+24x2

1 2

7 2

1

3 2

1 2

7 2

1

5 2

1 2

4

1

- 21 2

1 2

4

1

1 2

2

2

1 2

4

1

1

2

1 2

4

1

3 2

2

1 2

4

1

2

2

1 2

4

1

5 2

2

1 2

4

1

3

2

1

7 2

2

1 2

4



2 15y

3

1 24y



q √ √ x 105 x(tanh−1 1−y − tanh−1 x) −  48−231x+280x2 −105x3

1 2xy y 1−y

1 2

2

- y1



1 - 23 1

1

2

2 3y



1 - 23 1

3 2

2

1 - 4y





√

3 (1−y) 2

√1 x

2

2

1 - 43 1

1 4

2

 - y1 ln

− tanh−1 √ x −1

(3(1−y)−x) tanh

x
√

4(1−y)−x

1

2

+

√

x 1−y 3 (1−y) 2

tanh−1

3

− 2x 2

xy 1−y

1 15xy

1 3

3


√ 1−y 48(1−y)3 −231x(1−y)2 +280x2 (1−y)−105x3 (1−x−y)3

 24−60x−35x2 2 2 1 − 24(1−y) −60x(1−y)+35x 3 3 12y (1−x) 2 (1−x−y) 2
 √1−y 3(1−y)2 +100x(1−y)+105x2 2 1 − 3+100x+105x 64xy 1−x−y 1−x
√ x √ −(1+10x−35x2 ) tanh−1 x
 (1−y)2 +10x(1−y)−35x2 tanh−1 √ 1−y −3 x  6−7x 6(1−y)−7x  1 √ √ − 1−y−x 3y 1−x √
5 1 − y 3(1 − y)2 + 10x(1 − y) − 105x2 − (3 + 10x − 105x2 ) 384x2 y
√ x 2 3 −1 √
 x (1−y)3 +3x(1−y)2 +15x2 (1−y)−35x3 tanh−1 1−y −(1+3x+15x −35x ) tanh √ −3 x

1 - 23 1

1 - 32 1

5

(1−x−y) 2 15(1−y) −40x(1−y)+24x2  2

+ (1−x)3  3 2 2 (1−y)−35x3  1 16−72x+90x2 −35x3 − 16(1−y) −72x(1−y) +90x 5 5 8y 2 (1−x) 2
(1−x−y) √ x √ −1 √
 2 2  x (1−y−7x) tanh−1 1−y 81(1−y) −190x(1−y)+105x 1−y −(1−7x) tanh 81−190x+105x2 1 √ − + 15 - 48y 2 2 (1−x−y) (1−x) x

- 21 2

1

−

5(1−y)3 −30x(1−y)2 +40x2 (1−y)−16x3 

−

(1−x) 2 (1−x−y) 2  5−5y−6x 2 √ √ − 5y 1−x−y 1−x 9 2 −4 − (1 − y) 2 (1 − x − y)−4 ] y [(1 − x)

 5−6x

1 - 23 1

- 23

− x)− 2 − (1 − y)4 (1 − x − y)− 2 ]

1−y

3 (1−y) 2

1−x−y

√
x + ln

√ − (4 − x) 1 − x + 3 ln
√ x 1−y

3 (1−y) 2

2 3(1−y)2 +14x(1−y)−2x2 5 (1−y) 2

1−x−y 1−x





√

1−x−y

√
x − ln

1−x 1−x−y

√
 √ 1+ √1−x 1−y+ 1−x−y

− (3 + 6x − x2 ) tanh−1




√
x +

xy 1−y

− 4 ln

√ − 2(3 + 14x − 2x2 ) 1 − x + 30x ln

1−x 1−y−x




√
 √ 1+ √1−x 1−y+ 1−x−y

1 1 1 1 3 3 1

 4 4 x ) 4 − arctan(x 4 ) + x 4 (1 − y)− 4 ln( (1−y) 1 +x 1 ) − ln( 1+x 14 ) + 2 arctan( 1−y

 √ 2 2 1 2 3x 3 (1 − y)− 3 arctan - 2y 2




− (3 − x) tanh−1

3(1−y)2 +6x(1−y)−x2 tanh−1

 6y +

1−x 1−x−y

2

(1−y) 4 −x 4 √ 1 3x 3

1 2(1−y) 3

2

2

2

2(1−y) 3 +x 3

ln( 1−x−y 1−y ) 1 1
 1 3 3 ln( (1−y) −x ) − ln(1 − x 3 ) + 2 ln(1 − y) 1

1 +x 3

−(2 + x 3 ) ln(1 − x) − 3x 3 (1 − y)− 3

1−x 4 √ 1 3x 3

) − arctan( (1−y) 3

1 2+x 3


) +

2 (1−y) 3

6 σ

α1 α2

β 1 β2

1

- 12

1

1 2

2

1

- 12

1

1

2

1

- 12

1

3 2

2

1

- 12

1

2

2

1

- 13

1

2 3

2

F2 (σ; α1 , α2 ; β1 , β2 ; x, y) q  q x
 √ √
−1 x ( 1−y ) − x tanh−1 ( x) + ln 1−y−x - y1 2 1−y tanh 1−x √ q 1−x−y √
 2 √ 1+ √1−x − 1 − x + ln y 1−y 1−y+ 1−x−y √ x 
 √
(1−y+x) tanh−1 1−y 1−x √ − (1 + x) tanh−1 x − ln 1−x−y - y1 √1x 1−y √ √ 
 √ (1−y+2x) 1−x−y 1+ √1−x 2 √ y + − (1 + 2x) 1 − x + 3x ln √1−y+ 3xy 1−y 1−x−y
x 13 

1−x−y 1−x 1−x 1 − 13
− ln ln 1 1 1 3 y ln 1−x−y − 2y (1 − y) 3 √ 1 −2 3 (1 − y)− 3 arctan(

1

3 4

2

- y1

1 2

1

- 21

2

1 y

1

1 2

1

1

2

2 y

1

1 2

1

3 2

2

1

1 2

1

2

2

1

1

1

- 23

2

1

1

1

- 21

2

1

1

1

1 2

2

1

1

1

3 2

2

1

1

1

2

2 2

1

- 14

1

1

1

1

5 2

1

1

1

3

2

1

1

1

4

2

1

1

- 21

2

1

3 2 3 2

1

1 2

2

1

3 2

1

1

2

1

3 2

1

2

2

1

3 2

1

5 2

2

1

2

1

- 21

2

1

2

1

1 2

2

1

2

1

1

2

1

2

1

3 2

2

1

2

1

3

2

1

3

1

4

2

9 8

1

1

9 8

2





ln

ln

1−x−y 1−x




+x

1 4

1

1

2(1−y) 3 +x 3 − 41

(1 − y)



) − arctan(

ln

(1−y) 4 +x 4 1 (1−y) 4

√

1 3x 3 1 2+x 3

1

1

2xy 1−x 1−x−y − (1−x)(1−x−y) √
√ 1+ √1−x 1−y+ 1−x−y √ x √ √ tanh−1 x− 1−y tanh−1 1−y √ x




(1−x 3 )

(1−y) 3 −x 3 √ 1 3x 3

1 −x 4





 ) 1

− ln

1+x 4

1 1−x 4




−

x 1−y 1 4 (1−y)

2 arctan


14

1
 + 2 arctan(x 4 )

ln 

1 1−x + ln 1−x−y y 2 √ p 
 √ 1+ √1−x 2 1 − x + (1 − y)(1 − x − y) + x ln √1−y+ xy y − 1−x−y √ x  ln(1−y)  3 arcsin √ 1−y arcsin x
1−y−4x 2x 1−4x 2 − y 3x − − (1−x−y) 5 5 2 + (1−x)2 3y 2 2 (1−x−y) (1−x) √ √ x
  √ arcsin 1−y y - 2x x − arcsin 3x − ln(1−y) 3 y (1−x)(1−x−y) + y 2 2 (1−x−y) (1−x) √  √  arcsin √ x 2 x x 1−y √ √ − arcsin − ln(1−y) y y 1−x−y 1−x q √  ln(1−y) √ √ x √2 1 − x arcsin x − 1 − x − y arcsin 1−y − y xy   1 xy (1 − x − y) ln(1 − x − y) − (1 − y) ln(1 − y) − (1 − x) ln(1 − x) q   √
3 3 2 √1 x (1 − x − y) 2 arcsin 1−y − (1 − x) 2 arcsin x + y − ln(1−y) xy y x 
 1−x−y 1 2 2 2 − (1 − x) ln(1 − x) − x ln(1 − y) x2 y xy − (1 − x − y) ln 1−y 

 1−x−y 1 3 − (1 − x)3 ln(1 − x) − x3 ln(1 − y) + xy(5x + 2y − 4) 2x3 y 2 (1 − x − y) ln 1−y  4xy(x2 +y−1)
 1−x 1 y (1−x)2 (1−x−y)2 + ln 1−x−y 
 2xy 1 1−x y (1−x)(1−x−y) + ln 1−x−y √
 q 1−y 2 √1 √ 1+ √1−x y 1−x−y − 1−x + ln 1−y+ 1−x−y √ p
 √ 1+ √1−x 2 1 − x − (1 − y)(1 − x − y) − y + x ln √1−y+ xy 1−x−y √ x 3 √ 
−tanh−1 x
 (1−y) 2 tanh−1 1 1−x √ 1−y x ln − 2 y + xy 1−x−y
x √ x √   √ 5(1−y)−2x arcsin (5−2x) arcsin x
1−y 4−x − − + + ln(1−y) - xy 4(1−y)−x x 5 5 2 2 (1−x−y) (1−x) x (1−x−y) 2 (1−x) 2
√ x √
  3(1−y)−2x arcsin x y 1−y x √1 + − (3−2x) arcsin − ln(1−y) 3 3 y (1−x−y)(1−x) x x (1−x−y) 2 (1−x) 2 
 xy 1 1−x + ln y (1−x)(1−x−y) √1−x−y √ x   √ (1−y−2x) arcsin (1−2x) arcsin x 1−y √ √ + x ln(1 − y) - √1xy − 1−x−y 1−x  2

 1−x−y 1 1−x 2 − ln(1 − x) − xy x2 y x ln 1−x−y + (1 − y) ln 1−y 

 1−x−y 1 1−x 2 2 3 3 2x3 y 2x (1 − y) − 1 − x y + 2 (1 − y) ln( 1−y ) − ln(1 − x) + x ln( 1−x−y )
1 1 1
81  (1−y) 8 +x 8 (1−x 8 )
1
1 x
ln + 2 arctan 1−y − arctan(x 8 ) 1 1 1 1 (1−y) 8 −x 8 (1+x 8 ) x8 y
1
1 1
√ 1 1 √ √ 1 √ 1
√ 2 x(1−y) 8 (1−y) 4 − 2 x(1−y) 8 +x 4 (1+ 2x 8 +x 4 )  2x 8 − 12 + 2 arctan − arctan − 2 ln
1 1
√ 1 1 1 1 1 √ 1 (1−y) 4 −x 4

1−x 4

(1−y) 4 + 2 x(1−y)

8

+x 4 (1− 2x 8 +x 4 )

7 σ 7 6

α1 α2 β1 β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)
1 1 1 2  (1−y) 6 +x 6 (1−x 6 )2 7 1 1 1 6 2 ln
1 1 1 1 2 2x 6 y

5

1 1

13 6

2




7 (1−y) 6 −1 xy

1 1

6 5

2

6 5

1 1

11 5

2

5 4

3 4

1

1 2

2

5 4

3 4

1

3 2

2

5 4

1 1

5 4

2

5 4

1 1

9 4

2

4 3

1 1

4 3

2

4 3

11 8

1 1

7 3

2

1 1

11 8

2

−

7 7

6x 6 y

2

− arctan

q 1 1 1 (1+x 6 ) 1+x 6 +x 3 q 1 1 1 (1−x 6 ) 1−x 6 +x 3





1

1

1

1

1

1

1−x 3

1

+

√

(1−y) 6 +x 6 1

1

(1−y) 6 −x 6


r r




1

(1−y) 3 + x(1−y)


16

1

+x 3


1

1

1

(1−y) 3 − x(1−y) 6 +x 3
1 √ 3 x(1−y) 6

3 (1 − x − y) arctan

1 (1−y) 3

1 −x 3

− (1 − x) arctan

√

1

3x 6

1 1−x 3






1 2

1 2 √ √ 2 2 5+1 5−1 √ √ (1−y) 5 + (1−y) 5 − x(1−y) 5 +x 5
x(1−y) 5 +x 5
2 2



√ √ 5 + 1) ln 5 − 1) ln − ( ( 1 1 1 2 2 5+1 5−1 4x 5 y x 5 +x 5 x 5 +x 5 1+ 1− 2 2 √ √ √ 1 √ 1 p √ 10−2 5x 5 10−2 5x 5
+2 10 − 2 5 arctan( √ √ 1 1 ) − arctan( 1 ) 5 +( 5+1)x 5 4(1−y)√ 4+(√ 5+1)x 5 √ √ 1 1 1 1 p √ 10+2 5x 5 10+2 5x 5
(1−y) 5 −x 5
 +2 10 + 2 5 arctan( √ √ 1 ) − arctan( 1 1 ) − 4 ln 1 4(1−y) 5 −( 5−1)x 5 4−( 5−1)x 5 1−x 5
4 √ 1 √ 
5 6 (1−y) −1 2 2
− 36 ( 5 + 1) (1 − x − y) ln (1 − y) 5 + ( 5+1 ) x(1 − y) 5 + x 5 xy 2 5y 10x√ 1 2

5 5 −(1 − x) ln 1 + ( 5+1 2 )x + x √ √ √
15 2 2

2
1 5−1 5 5 5 −( 5 − 1) (1 − x − y) ln (1 − y) 5 − (√ 5−1 − (1 − x) ln 1 − ( + x ) x(1 − y) 2 √ 1 2 )x + x √ √ 1 p √
5 5 10−2 5x 10−2 5x +2 10 − 2 5 (1 − x − y) arctan( √ √ 1 1 ) − (1 − x) arctan( 1 ) 5 +( 5+1)x 5 4(1−y) 4+( 5+1)x 5 √ √ √ 1 √ 1
p √ 10+2 5x 5 10+2 5x 5 +2 10 + 2 5 (1 − x − y) arctan( √ √ 1 1 ) − (1 − x) arctan( 1 ) 4(1−y) 5 −( 5−1)x 5 4−( 5−1)x 5 1 1
1
 −4 (1 − x − y) ln (1 − y) 5 − x 5 − (1 − x) ln(1 − x 5 ) √ √ √ √  √√ 1−y+ 1−x−y 1+ 1−x  2 2 √ √ − y 1−x−y 1−x √  √ √ √ 1 32 − 21 1 − y + 1 − x − y) − (1 + 1 − x)− 2 ( y
1 1 1
41  (1−y) 4 +x 4 (1−x 4 )
1
 x 1
ln + 2 arctan 1−y − arctan x 4 1 1 1 1 1

(1−y) 4 −x 4 (1+x 4 )

x4 y

1 1 1
1 3 4 4 4x 4 (1 − y) 4 − 1 + (1 − x) ln 1+x 41 − (1 − x − y) ln (1−y) 1 +x 1 y 1−x 4 (1−y) 4 −x 4
14
 1 x 4 − (1 − x − y) arctan +2 (1 − x) arctan x 1−y
13 2
1 2 2 √ 1
 √ 1 √  +x 3 (1−x 3 )
(1−y) 3 + x(1−y) 1 3x 3 3x 3
ln + 2 3 arctan 1 − arctan 1 1 1 2 1 1 2 1

5

5 4x 4



2x 3 y 4

2(1−y) 3 +x 3 r

(1+x 3 +x 3 ) (1−y) 3 −x 3



1 3

2 3

q 2 1 1+x 3 +x 3




2+x 3

2

(1−y) 3 + x(1−y)

− (1 − x − y) ln 3x (1 − y) − 1 + (1 − x) ln 1 1 1 y 1−x 3 (1−y) 3 −x 3 √ 1 √ 1 √
 3x 3 + 3 (1 − x) arctan 3x 13 − (1 − x − y) arctan 1 1 2+x 3 2(1−y) 3 +x 3

18 1
√ 1 1 √ 1 1 1 1  +x 4 (1− 2x 8 +x 4 ) (1−y) 8 +x 8 (1−x 8 ) (1−y) 4 + 2 x(1−y) 1 1
ln − 2− 2 ln
1
√ 1 1 1 3 √

4 3x 3

1

(1−y) 8 −x 8 (1+x 8 )

x8 y

x(1−y)


81

1 −x 4

(1−y) 4 − 2 x(1−y)

− arctan

√ 1 2x 8 1 1−x 4




8

1

1

1

+x 4 (1+ 2x 8 +x 4 ) 1

1

x ) 8 − arctan x 8 − 2 arctan( 1−y

1 √ √ 2 5 5+1 5−1 1





2 2 √ (1−y) 5 + 2 +x 5 1− 2 x 5 +x 5 x(1−y)
ln 5 ln

1 2
2 1 √ √ 1 5 − 2 1 2 4x 3 y (1−x) (1−y) 5 −x 5 1+ 5+1 x 5 +x 5 x(1−y) 5 +x 5 (1−y) 5 − 5−1 2 2 √ √ √ 1 √ 1 p √ 10+2 5x 5 10+2 5x 5
+2 10 − 2 5 arctan √ √ 1 1 − arctan 1 5 −( 5−1)x 5 4(1−y)√ 4−(√ 5−1)x 5 √ 1 √ 1
p √ 10−2 5x 5 10−2 5x 5 −2 10 + 2 5 arctan √ √ 1 1 − arctan 1 1

1 1

1

 (1 − x − y) ln

1 (1−y) 4

7 5

1

1

(1−y) 3 −x 3

√ √ 2 + 2 arctan

7 5

1

+x 3 (1−x 6 +x 3 )


16

1

−(1 − x) ln 6 5


16

(1−y) 6 −x 6 (1+x 6 )2 (1−y) 3 − x(1−y) +x 3 (1+x 6 +x 3 )
16 √ √ 1
 3 x(1−y) 3x 6

√ +2 3 arctan

7 6

1

(1−y) 3 + x(1−y)

1



(1−x−y)(1−x 5 )5

4(1−y) 5 +( 5+1)x 5

4+( 5+1)x 5


31

2

+x 3

8 σ α1 α2 β1 β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)
3  7 (1−y) 5 −1 7 12 − 77 (1 − x − y) ln 1 1 2 5 5 2xy

1−x−y 1

20x 5 y

√

2 √ (1−y) 5 + 5+1 2 − 5 (1 − x − y) ln √ 2

3 2

- 21

1

1 2

2

3 2

- 21 1

3 2

2

3 2

- 21 1

5 2

2

3 2

- 21 1

7 2

2

3 2

1 2

1

- 21

2

3 2

1 2

1

1 2

2


51


5 − (1 − x) ln

1−x 1

(1−x 5 )5 √

2 1 2 +x 5 x 5 +x 5
1+ 5+1 2 − (1 − x) ln √5−1
2 1 1 2 1− 2 x 5 +x 5 (1−y) 5 − 5−1 x(1−y) 5 +x 5 2 √ √ √ √ 1 1 p √ 10+2 5x 5 10+2 5x 5
+2 10 − 2 5 (1 − x − y) arctan √ √ 1 − (1 − x) arctan 1 1 5 −( 5−1)x 5 4(1−y)√ 4−(√ 5−1)x 5 √ 1 √ 1
p √ 10−2 5x 5 10−2 5x 5 −2 10 + 2 5 (1 − x − y) arctan √ √ 1 1 − (1 − x) arctan 1

x(1−y)

4(1−y) 5 +( 5+1)x 5 4+( 5+1)x 5 1 1 2 −1 2 2 (1 − x − y) − (1 − x) ] y [(1 − y) √ x √ √   √ arcsin x 1−x−y 1−y −arcsin 1 √ − 1−x + y 1−y x √ x √  (1−y+2x)√1−x−y √ x (1−y−4x) arcsin 1−y −(1−4x) arcsin 3 √ − (1 + 2x) 1 − x − 8xy 1−y x

√ √ x √ 2  x
(1−y)2 −4x(1−y)+8x2 arcsin 1−x−y 3(1−y)2 −10x(1−y)−8x2 1−y −(1−4x+8x ) arcsin 5 √ − 64x2 y 3 1−y x  √ 2

+(3 − 10x − 8x ) 1 − x  

1−y−2x 2 1−2x y (1−x−y) 32 − (1−x) 23 2 − 21 − (1 y [(1 − x − y)

3 2

1 2

1

3 2

2

√2 xy

3 2

1 2

1

5 2

2

3 2xy

3 2

1 2

1

7 2

2

32x2 y

3 2 3 2

1

1

- 21 2

1

1

1 2

2

3 2

1

1

3 2

2

3 2

1

1

2

2

3 2

1

1

5 2

2

3 2

1

1

3

2

3 2

1

1

7 2

2

3 2

1

1

4

2

3 2

3 2

1

- 21 2

3 2

3 2

1

1 2

2

3 2

3 2

1

5 2

2

3 2

3 2

1

7 2

3 2 3 2 3 2

2

1

- 21 2

2

1

1 2

2

2

1

1

2

3 2

2

1

3 2

2

3 2

2

1

5 2

2

3 2

2

1

3

2

2

1

(1−y) 5 −x 5



arcsin

q

x 1−y

1

− x)− 2 ] √  − arcsin x

√ x √ √ √ x (1−y−2x) arcsin 1−y −(1−2x) arcsin √ 1−x−y− 1−x− x
√ x  3(1−y)2 −8x(1−y)+8x2 arcsin √ 1−y −(3−8x+8x2 ) arcsin x 15 √

x
 √ √ −3 (1 − y − 2x) 1 − x − y − (1 − 2x) 1 − x 1   1−3x 2 (1−y) 2 (1−y−3x) − (1−x) 2 y (1−y−x)2 1

2 y [(1

− y) 2 (1 − x − y)−1 − (1 − x)−1 ] q  √  x √2 arctan 1−y − arctan x xy √  √ √ 4 1−x+ 1−y− 1−x−y−1 xy √ x −1 √   x (1−x−y) tanh−1 1−y −(1−x) tanh 3 √ √ 1 − y − 1 − xy x   3 3
1 8 2 2 − (2 − 2y − 3x)(1 − y) 2 + 2 − 3x 3x2 y 2 (1 − x − y) − (1 − x) √ x  (1−x−y)2 tanh−1 √ 1−y
 √ −(1−x)2 tanh−1 x
5 √ − 1 − y 3(1 − y) − 5x + 3 − 5x 3 2 4x y x  5 5 5
3
- 5x43 y 8 1 − (1 − y) 2 − (1 − x) 2 + (1 − x − y) 2 − 20x 1 − (1 − y) 2 3 1
 −20x(1 − (1 − y) 2 ) + 15x2 1 − (1 − y) 2   2 (1−y−4x)(1−y) − 1−4x5 5 y (1−x−y) 2 (1−x) 2 3 2 − 23 − (1 − x)− 2 ] y [(1 − y)(1 − x − y) √ √ x  √ 3 (1−y) arcsin √1−y −arcsin x − 1− xy x

x−y+

 √ 1−x


√ √ 3(1 − y) − 2x 1 − x − y − (3 − 2x) 1 − x − 3   2 (1−y−5x)(1−y) 2 1−5x − (1−x) 3 y (1−x−y)3 15 8x2 y



3 2 −2 2 y [(1 − y) (1 − x − y)   2−x 1 2(1−y)−x y (1−x−y) 32 − (1−x) 23

− (1 − x)−2 ] √

√ x √ x √ x   (1−y+x) tanh−1 √ 1−y √ −(1+x) tanh−1 x 3 √ − 1−y+1 2xy x
 √ √ 3 8 2 3x2 y 2 (1 − y) − 1 + (2 + x) 1 − x + 2(1 − y) + x) 1

 √1−y 1 y 1−x−y

−

1 1−x

+

tanh−1




(1−y) 3(1−y)−4x arcsin √

−1 x 1−y −tanh

−x−y



√ x

√ x x 1−y −(3−4x) arcsin



9 σ 3 2 3 2 3 2 3 2 3 2

α1 α2 β1 β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)
√ x −1 √ 
 (1−x−y) 3(1−y)+x tanh−1 x 1−y −(1−x)(3+x) tanh 7 15 √ √ 2 1 2 1 − y 3(1 − y) − x − (3 − x) − 2 8x2 y x
 
3 3 3 2 1 4 2 5x163 y 4(1 − y) + x (1 − x − y) 2 − 4(1 − y) − 5x (1 − y) 2 − (4 + x)(1 − x) 2 + 4 − 5x   2 5 − 1−6x7 1 - 21 2 y2 (1−y−6x)(1−y) 7 2 5 2 5 2

2

1

1 2 3 2

1

3 2

5 2

1

7 2

2

3 2 3 2

3

1

- 21 2

3

1

1 2

2

3 2

3

1

1

2 2

2

3 2

3

1

3 2

3 2

3

1

2

2

3 2

3

1

5 2

2 2

3 2

3

1

7 2

3 2

3

1

4

2

3 2

7 2

1

- 21

2

3 2

7 2

1

1 2

2

3 2

7 2

1

3 2

2

3 2

7 2

1

5 2

2

3 2 3 2

4

1

- 21 2

4

1

1 2

2

3 2

4

1

1

2

3 2

4

1

3 2

2

3 2

4

1

2

2

3 2

4

1

5 2

2

3 2

4

1

3

2

1

7 2

2

2

(1−x−y) 2 (1−x) 2 2 2 − 52 − y [(1 − y) (1 − x − y)   2 3(1−y)−2x 3−2x 3y (1−x−y) 32 − (1−x) 32 5 4x2 y



3

(1−y)2 arcsin

√

x

5

(1 − x)− 2 ]

−arcsin

√ 1−y x

5

√

x


√ √ − 3(1 − y) + 2x 1 − x − y + (3 + 2x) 1 − x

  2 (1−y−7x)(1−y) 2 1−7x − (1−x) 4 y (1−x−y)4 5 2 −3 2 − (1 − x)−3 ] y [(1 − y) (1 − x − y)  2 2 2 1 8(1−y) −8x(1−y)+3x − 8−8x+3x 5 5 4y (1−x−y) 2 (1−x) 2  √1−y




5(1−y)−3x 5−3x − (1−x) 2 (1−x−y)2   4−3x 1 4(1−y)−3x 2y (1−x−y) 32 − (1−x) 32 1 4y

 (1−y+3x) tanh−1 3 8xy 15 32x2 y



√

+3

tanh−1

−1 x 1−y −(1+3x) tanh

√

x




√

3(1−y)2 +2x(1−y)+3x2 tanh−1 √ x

√

√

x

−1 x 1−y −tanh

√ x

−

√ x

√ 1−y(1−y−3x) 1−x−y

2 −1 x 1−y −(3+2x+3x ) tanh

−

√

x

1−3x 1−x



√


 1 − y(1 + x + y) − (1 + x) 
 √ √ 5 4 2 2 2 2 2 5x2 y 8 (1 − y) − 1 + (8 + 4x + 3x ) 1 − x − (8(1 − y) + 4x(1 − y) + 3x ) 1 − x − y   3 2 (1−y−8x)(1−y) − 1−8x9 9 y (1−x−y) 2

2 y [(1

−3

(1−x) 2 7

7

− y)3 (1 − x − y)− 2 − (1 − x)− 2 ]

 15(1−y)2 −20x(1−y)+8x2 2 5 15y (1−x−y) 2   2 5(1−y)−4x 5−4x 5y (1−x−y) 32 − (1−x) 32

−

15−20x+8x2 5 (1−x) 2



7

3 2

4

8 5

1

1

8 5

13 8

1

1

13 8

2

  2 (1−y−9x)(1−y) 2 1−9x − (1−x) 5 y (1−x−y)5 7 2 −4 2 − (1 − x)−4 ] y [(1 − y) (1 − x − y)  3 2 2 3 2 3 1 16(1−y) −24x(1−y) +18x (1−y)−5x − 16−24x+18x7 −5x 7 8y (1−x−y) 2 (1−x) 2
√ x −1 √  2 2 √  x tanh−1 33(1−y) −40x(1−y)+15x 1−y 1−y −tanh 1 33−40x+15x2 √ − + 15 3 3 24y (1−x−y) (1−x) x  2 2 2 1 8(1−y) −12x(1−y)+5x − 8−12x+5x 5 5 4y (1−x−y) 2 (1−x) 2
√ x √ √ −1  −(1+5x) tanh−1 x 2 (1−y+5x) tanh 1−y 3(1−y)2 −22x(1−y)+15x2 1−y 1 √ − + 3−22x+15x 16xy 3 (1−x−y)2 (1−x)2 x   1 6(1−y)−5x 6−5x 3y (1−x−y) 32 − (1−x) 32


√

 (1−y)2 +2x(1−y)+5x2 tanh−1 5 √ 2 64x y 3 x  3+4x−15x2 + 1−x 1

3 4x 5

y

√ ( 5 + 1) ln

2

2 −1 x 1−y −(1+2x+5x ) tanh

√ 5−1 x(1−y) 2 √ 2 5−1 1 1− 2 x 5 +x 5

(1−y) 5 −


51

2

+x 5




√ x

√ − ( 5 − 1) ln √ √ 1

−

√

1−y 3(1−y)2 +4x(1−y)−15x2 1−x−y

2

√ 5+1 x(1−y) 2 √ 2 5+1 1 1+ 2 x 5 +x 5

(1−y) 5 +


51

√ √ 1 1
1 p √ 5 5 10−2 5x 5 10−2 5x 5
− arctan + 2 −4 ln (1−y) −x 10 + 2 5 arctan √ √ 1 1 1 1 1−x 5 4(1−y) 5 +( √ 5+1)x 5 4+( 5+1)x 5 √ √ 1 √ 1
 p √ 10+2 5x 5 10+2 5x 5 −2 10 − 2 5 arctan √ √ 1 1 − arctan 1 4(1−y) 5 −( 5−1)x 5 4−( 5−1)x 5

√ 1 1 √ 1 1 1 1 1 1  (1−y) 4 + 2(x(1−y)) 8 +x 4 (1− 2x 8 +x 4 ) (1−y) 8 +x 8 (1−x 8 ) 1 1

√ 1 1 − 2− 2 ln ln √ 5 1 1 1 1 1 1 x8 y (1−y) 8 −x 8 (1+x 8 ) (1−y) 4 − 2(x(1−y)) 8 +x 4 (1+ 2x 8 +x 4 )
1 √ √ 1
 √ 2 x(1−y) 8 1
1 x +2 arctan( 1−y ) 8 − arctan x 8 − 2 arctan − arctan 2x 18 1 1 (1−y) 4 −x 4

1−x 4

2

+x 5







10 σ

α1 α2

β1 β2

5 3

1

1

5 3

2

5 3

1

1

8 3

2

7 4

1 4

1

1 2

2

7 4

1 4

1

3 2

2

7 4

1

1

7 4

2

7 4

1

1

11 4

2

7 4 7 4

5 4 5 4

1

2

1

1 2 3 2

9 5

1

1

9 5

2

9 5

1

1

14 5

2

11 6

1

1

11 6

2

2

F2 (σ; α1 , α2 ; β1 , β2 ; x, y) 1 3 √ 1 √ 1
 √  3) 1 3x 3 3x 3
ln (1−x−y)(1−x 3 arctan 1 − arctan 2 1 1 1 3 − 2 1 3 3 3 2x y 2(1−y) +x 2+x 3 (1−x) (1−y) 3 −x 3
1  5 (1−y) 3 −1 1−x−y 1−x
− 55 (1 − x − y) ln 1 1 1 3 − (1 − x) ln 2xy 3 6x 3 y

2x 6 y

(1−y) 3 − x(1−y) 6 +x 3
1 √ 3 x(1−y) 6

√ −2 3 arctan 1

11 6

1

1

17 6

2




11 (1−y) 6 −1 5xy

−(1 − x) ln

−

1

1

15 8

2

2

- 21

1

1 2

2

2

- 21

1

1

2

2

- 21

1

3 2

2

2

- 21

1

2

2

2

- 21

1

5 2

2

1 7

x8 y



ln

1

1

(1−y) 3 −x 3 11

6x

11 6

y



1

√ 1 3x 6 1

1−x 3

r

(1 − x − y) ln r



1

1

(1−y) 8 −x 8 (1+x 8 )




1

(1−y) 3 + x(1−y) 1

(1−y) 3 − x(1−y)

−


61


16

1

+ 2− 2 ln

1

1

1

1

1

1

+x 3 (1−y) 6 +x 6 +x 3 (1−y) 6 −x 6
1 √ 3 x(1−y) 6

√ 3 (1 − x − y) arctan

1

1

(1−y) 8 +x 8 (1−x 8 )

(1+x 6 +x 3 )(1+x 6 )2

(1−y) 6 −x 6

− arctan

q 1 1 1 1+x 6 +x 3 (1+x 6 ) q 1 1 1 1−x 6 +x 3 (1−x 6 ) 1

15 8

(1−x 3 )

(1−y) 3 −x 3

√ 1
 √ 1 √ 3x 3 3x 3 −2 3 (1 − x − y) arctan 1 − (1 − x) arctan 1 1 2(1−y) 3 +x 3 2+x 3 √ √ √ √  √√ 1+ 1−x  2 2 √ 1−y+ 1−x−y − √1−x 3y (1−x−y)(1−y) p  √√1−y+√x−√√1−y−√x p √ √  √4 √ − 1+ x+ 1− x 3 xy 1−y
1 1 1  (1−y) 4 +x 4 (1−x 4 ) 1 1
 1 x
ln ) 4 − arctan x 4 − 2 arctan( 1−y 3 1 1 1 x4 y (1−y) 4 −x 4 (1+x 4 )
1 1 1 1  7 (1−y) 4 −1 4 4 − 77 (1 − x − y) ln (1−y) 1 +x 1 − (1 − x) ln 1+x 41 3xy 4x 4 y (1−y) 4 −x 4 1−x 4 1 1
 x ) 4 − (1 − x) arctan x 4 −2 (1 − x − y) arctan( 1−y √ √ √ −3 √ √ √ 3 3 3 2 x) 2 + ( 1 − y + x)− 2 − (1 − x)− 2 − (1 + x)− 2 ] 3y ( 1 − y − √ √ √ √ √ √ 1 1 1 1 √4 ( 1 − y − x)− 2 − ( 1 − y + x)− 2 − (1 − x)− 2 + (1 + x)− 2 ] 3 xy
√ √ 2 2 1 1 2 1 √  (x(1−y)) 5 +x 5 (1+ 5+1 x 5 +x 5 ) (1−y) 5 − 5−1 (1−x−y)(1−x 5 )5 2 2 1

√ √ ln 5 ln 2 2 2 4 1 1 1 5 − 1 (1−y) 5 + 5+1 4x 5 y (x(1−y)) 5 +x 5 (1− 5−1 x 5 +x 5 ) (1−x) (1−y) 5 −x 5 2 2 √ √ √ √ 1 1 p √ 10+2 5x 5 10+2 5x 5
−2 10 + 2 5 arctan √ √ 1 1 − arctan 1 5 −( 5−1)x 5 5 4(1−y)√ 4−( √ 5−1)x √ 1 √ 1
 p √ 10−2 5x 5 10−2 5x 5 −2 10 − 2 5 arctan √ √ 1 1 − arctan 1 4(1−y) 5 +( 5+1)x 5 4+( 5+1)x 5
√ √ 1 2 2 1 1 2 √ 9 (1−y) 5 −1 (x(1−y)) 5 +x 5 1− 5−1 x 5 +x 5
(1−y) 5 − 5−1 9 2 2 √ √ + − (1 − x) ln 5 (1 − x − y) ln 2 2 9 2 1 1 4xy 20x 5 y (1−y) 5 + 5+1 1+ 5+1 (x(1−y)) 5 +x 5 x 5 +x 5 2 2
1−x−y 1−x
− (1 − x) ln − (1 − x − y) ln 1 1 1 5 5 )5 (1−y) 5 −x 5 √ (1−x √ √ 1 √ 1 p √ 10+2 5x 5 10+2 5x 5
+2 10 + 2 5 (1 − x − y) arctan − (1 − x) arctan √ √ 1 1 1 5 −( 5−1)x 5 5 4(1−y)√ 4−( √ 5−1)x √ 1 √ 1
 p √ 10−2 5x 5 10−2 5x 5 +2 10 − 2 5 (1 − x − y) arctan √ √ 1 1 − (1 − x) arctan 1 4(1−y) 5 +( 5+1)x 5 4+( 5+1)x 5


1 1 1 1 1 2 1 1 1  (1−y) 6 +x 6 (1−x 6 +x 3 )(1−x 6 )2 (1−y) 3 + x(1−y) 6 +x 3 1 ln


5 1 1 1 1 1 1 2 1 1

1 (1−y) 3

1 (1−y) 4

1 −x 4

(1−y) 2

3

1

1

− y)− 2 (1 − x − y) 2 − (1 − x) 2 ] √ x  y √
 (1−x−y) tanh−1 1−y 1 √1 + − (1 − x) tanh−1 x 3 2y 1−y x (1−y) 2 
23  3 1−x−y 2 2 3xy (1 − x) − 1−y √ x  xy √
 (1−x−y)2 tanh−1 1−y 3 √1 − (1 − x)2 tanh−1 x − 3 8xy 1−y x (1−y) 2




− (1 − x) arctan


1 1
√ 1 1 √ 1 (1−y) 4 + 2 x(1−y) 8 +x 4 (1− 2x 8 +x 4 )
1 1
√ 1 1 √ 1 (1−y) 4 − 2 x(1−y) 8 +x 4 (1+ 2x 8 +x 4 )
1 √ √ 1 √ 2 x(1−y) 8 2x 8

1 1
x ) 8 − arctan x 8 − 2 arctan −2 arctan( 1−y √ x  y √ tanh−1 1−y √
 1 − − tanh−1 x x 3 y 1−y

1 y [(1

1 −x 3




− arctan

1 1−x 4




√

1

3x 6

1 1−x 3




11 σ α1 α2 β1 β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)  (1−x−y) 25  5 8 2 - 12 1 3 2 − (1 − x) 2 + y 3 2 15x y 2 (1−y) x  3 (1−x−y)3 tanh−1 √ 1−y √
1 7 5 √ 2 -2 1 2 − (1 − x)3 tanh−1 x − 3 2 48x2 y x (1−y) 2

7

2 - 12 1

4

2

8 35x3 y

1 8

9 8

2

1

2

1



(1−x−y) 2


2 (1 − y)2 − 1 + 7xy − 2

3 (1−y) 2

1 6

1

7 6

2

1

1 12x 6

y



5

(1 − y)− 6 ln

1 (1−y) 6

1 +x 6

1 (1−y) 6

1 −x 6

2

1

2

2

1 5

1

6 5

2

2

1 5

1

11 5

2

2

1 4

1

5 4

2

2

1 4

1

9 4

2

2

1 3

1

4 3

2

-

 1−x−y

7 7

5

36x 6 y (1−y) 6

1 (1−y) 6


2

1 (1−y) 3 1

(1−y) 6 +x 6

ln


2

1 (1−y) 3 1 (1−y) 3 1 6


√ 3 x(1−y)

1

13 6






1 1 1
81  7 7 1
8 8 x (1 − y)− 8 ln (1−y) 1 +x 1 − ln 1+x 81 + 2 (1 − y)− 8 arctan 1−y − arctan(x 8 ) y (1−y) 8 −x 8 1−x 8
81 √ √ 1
√ 2 x(1−y) 7 −8 − arctan 2x 18 + 2 (1 − y) arctan 1 1 4 (1−y) 4 −x 4
81 1 1−x √ 1 √ 1 1  4 4 +x − (1−y) 2 x(1−y) 7 1 1− 2x 8 +x 4 −2− 2 (1 − y)− 8 ln
1 1 − ln √ 1 1 √ 1

√ 5 +2 3 (1 − y)− 6 arctan 1 6

+ 3 − 8x − 3x2

1 8x 8

8

(1−y) 4 + 2 x(1−y)

2

7

− (1 − x) 2

3(1−y)2 −8x(1−y)−3x2 1−y

1 −x 6


r

+ x(1−y)

− arctan

1 −x 3 1

1 (1−y) 3


1 √ 3 x(1−y) 6


61

− x(1−y)

(1−y) 3 + x(1−y) r




1+ 2x 8 +x 4

+x 4


16 √

1 +x 3 1

3x 6

1 1−x 3


61

− x(1−y)

1

+x 3


16

1

+x 3 1 +x 3




1 6

1 6 2

1 3

(1+x ) (1+x +x )
− ln 1 1 1 6 2 6 3

(1−x ) (1−x +x )




q 1 1 1+x 6 +x 3 q 1 1 1 (1−x 6 ) 1−x 6 +x 3 1




− (1 − x) ln

(1+x 6 )




√ 1

 √
− (1 − x) arctan 3x 16 + 3 1−x−y5 arctan 1 1 6 3 3 (1−y) (1−y) −x 1−x 3 √ √ √

1
1  4 2 2

2
1 x(1 − y) 5 + x 5 − ln 1 + 5+1 x5 + x5 ( 5 + 1) (1 − y)− 5 ln (1 − y) 5 + 5+1 1 2 2 20x 5 y √ √ √

1
1 2 2

2
4 5−1 x(1 − y) 5 + x 5 −√ x5 + x5 ln 1 − 5−1 −( 5 − 1) (1 − y)− 5 ln (1 − y) 5 − √ 2 2 √ √ 1 1 p √ 4 10−2 5x 5 10−2 5x 5
+2 10 − 2 5 (1 − y)− 5 arctan( √ √ 1 ) − arctan( 1 1 ) 5 +( 5+1)x 5 4(1−y)√ 4+(√ 5+1)x 5 √ 1 √ 1
p √ 4 10+2 5x 5 10+2 5x 5 +2 10 + 2 5 (1 − y)− 5 arctan( √ √ 1 1 ) − arctan( 1 ) 4(1−y) 5 −( 5−1)x 5 4−( 5−1)x 5 1 1
1
 4 −4 (1 − y)− 5 ln (1 − y) 5 − x 5 − ln(1 − x 5 ) √ √
1  2 2
) x(1 − y) 5 + x 5 - 36 ( 5 + 1) ( 1−x−y4 ) ln (1 − y) 5 + ( 5+1 2 50x 5 y √(1−y) 5 1 2

5 5 −(1 − x) ln 1 + ( 5+1 2 )x + x √ √ √
15 1 2 2
2

1−x−y 5−1 5−1 5 − ( 5 ) x(1 − y) )x 5 + x 5 − (1 − x) ln 1 − ( −( 5 − 1) ( + x 4 ) ln (1 − y) 2 2 5 (1−y) √ √ √ 1 √ 1 p √ 10−2 5x 5 10−2 5x 5
) − (1 − x) arctan( +2 10 − 2 5 ( 1−x−y4 ) arctan( √ √ 1 1 1 ) 5 +( 5+1)x 5 (1−y) 5 4(1−y) 4+( 5+1)x 5 √ √ √ √ 1 1 p √ 10+2 5x 5 10+2 5x 5
+2 10 + 2 5 ( 1−x−y4 ) arctan( √ √ 1 1 ) − (1 − x) arctan( 1 ) (1−y) 5 4(1−y) 5 −( 5−1)x 5 4−( 5−1)x 5 1
1
 1 1−x−y 5 − x5 −4 ( − (1 − x) ln(1 − x 5 ) 4 ) ln (1 − y) (1−y) 5

1

1 4x 4

y

5 5 16x 4

y

1 1 6x 3

 3 (1 − y)− 4 ln

y



1

1

1 (1−y) 4

1 −x 4

(1−y) 4 +x 4 1

(1 − x) ln 1+x 41 − 1−x 4

2

1

7 3

2

1 1−x 4 1 4




3

(1−y) 3 +

1 4

ln (1−y) 1 +x 1 + 2 (1 − x) arctan x 4 − (1−y) 4 −x 4
31 2 1 2
+x 3
x(1−y) 3
− ln 1+x 3 +x 1 1 2 1 2 1

(1−y) 3 −x 3 √ 1 3x 3 1

1

2(1−y) 3 +x 3

− arctan r

(1−x 3 ) √ 1
 3x 3 1

2+x 3


1 2 2 2 1  (1−y) 3 + x(1−y) 3 +x 3 1+x 3 +x 3 1−x−y − (1 − x) ln 1 4 1 2 ln 1 9x 3 y 1−x 3 (1−y) 3 (1−y) 3 −x 3 √ 1 √ 1 √
 1−x−y 3x 3 3x 3 + 3 (1 − x) arctan 1 − 2 arctan 1 1 4

q

2+x 3

x 1−y

+ 2 (1 − y)− 4 arctan

3 (1−y) 4

√ 2 +2 3 (1 − y)− 3 arctan

1 3

1

1+x 4

− ln

1−x−y 2

 2 (1 − y)− 3 ln




(1−y) 3

2(1−y) 3 +x 3


41

1

− arctan x 4

1−x−y 3 (1−y) 4

arctan




x 1−y


41


12 σ α1 α2 β1 β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y) 1 1 1  (1−y) 8 +x 8 3 1+x 8 − 58 (1 − y) 2 38 1 11 2 ln 1 − ln 3 1 1 8 8 8 8 8 8x y (1−y) −x 1−x
1 1 √ 1 √ 1 1
(1−y) 4 + 2 x(1−y) 8 +x 4 1 5 1+ 2x 8 +x 4 −2− 2 (1 − y)− 8 ln
1 1 − ln √ 1 1 √ 1 (1−y) 4 − 2 x(1−y)
1 √ 2 x(1−y) 8

2

2 5

1

7 5

2

2

2 5

1

12 5

2

2

1

- 21 2

2

1 2 1 2

1

1 2

2

1 2

1

3 2

2

1 2

1

2 2

2

1 2

1

5 2

2

1 2

1

3 2

2

1 2

1

7 2

2

2

3 5

1

8 5

2

2

2

5 8

1

13 8

2

1

1

1 y [(1

2x 2 y

 (1−x−y) 2 8 √ 3x2 y 1−y

3

− (1 − x) 2 + y √ x 2 −1

5

5

8x 8 y

2

2 3

5 3

1

8 3

2

− 21

− 38

(1 − y)

2

1

1 2 3x 3

-

y

5 5 9x 3

1−y

− (1 − x)2 tanh−1

4(1−y) 5 −( 5−1)x 5

1 1 1  3 8 8 (1 − y)− 8 ln (1−y) 1 +x 1 − ln 1+x 81

+2 (1 − y) 1



 √
x +y
1 2 √ 2 √ √ (1−y) 5 − 5−1 x(1−y) 5 +x 5
1 2
2 3 − 52 5 5 − ln(1 − 5−1 ( 5 + 1) (1 − y) ln 3 2 2 x +x ) 20x 5 y (1−y) 5
1 2 √ 2 √ √ (1−y) 5 + 5+1 x(1−y) 5 +x 5
1 2 2
2 x5 + x5 ) − ln(1 + 5+1 −( 5 − 1) (1 − y)− 5 ln 2 2 (1−y) 5 1 1 1
(1−y) 5 −x 5
− 25 − ln(1 − x 5 ) −4 (1 − y) ln 1 (1−y) 5 √ √ √ 1 √ 1 p √ 2 10−2 5x 5 10−2 5x 5
+2 10 + 2 5 (1 − y)− 5 arctan √ √ 1 1 − arctan 1 5 +( 5+1)x 5 4(1−y)√ 4+(√ 5+1)x 5 √ √ 1 1 p √ 2 10+2 5x 5 10+2 5x 5
 −2 10 − 2 5 (1 − y)− 5 arctan √ √ 1 1 − arctan 1  1 (1−x−y) tanh 15 √ √ 8x2 y x 1−y

− 83

2 3

1

− y)− 2 (1 − x − y)− 2 − (1 − x)− 2 ] x  arctan √ 1−y √  √1 √ − arctan x xy 1−y q √  2 1 − x − 1−x−y xy 1−y x  (1−x−y) tanh−1 √ 1−y √  √ - 33 − (1 − x) tanh−1 x 1−y

−2

2

1− 2x 8 +x 4

+x 4

√ 1
√ 5 1
 1 5 x + 2 (1 − y)− 8 arctan − arctan 2x 18 − 2 (1 − y)− 8 arctan( 1−y ) 8 − arctan x 8 1 1 (1−y) 4 −x 4 1−x 4  1−x−y 1 1−x − 53
(1 − y) − ln ln 2 1 1 1 5 10x 3 y (1−x 5 )5 (1−y) 5 −x 5
1 √ √ 2 2 2 1 √ (1−y) 5 + 5+1 x(1−y) 5 +x 5 3 x 5 +x 5
1+ 5+1 2 2 √ − 5 (1 − y)− 5 ln − ln
2 1 1 √ 5−1 2 2 1− 2 x 5 +x 5 (1−y) 5 − 5−1 x(1−y) 5 +x 5 2 √ √ √ √ 1 1 p √ 3 10+2 5x 5 10+2 5x 5
+2 10 − 2 5 (1 − y)− 5 arctan √ √ 1 1 − arctan 1 5 −( 5−1)x 5 5 4(1−y)√ 4−( √ 5−1)x √ 1 √ 1
 p √ 3 5 5 10−2 5x 10−2 5x −2 10 + 2 5 (1 − y)− 5 arctan √ √ 1 1 − arctan 1 4(1−y) 5 +( 5+1)x 5 4+( 5+1)x 5  1−x−y 1−x
- 77 1−x−y3 ln 1 1 1 5 − (1 − x) ln 50x 5 y (1−y) 5 (1−x 5 )5 (1−y) 5 −x 5
1 √ √ 2 2 2 1 √ (1−y) 5 + 5+1 x(1−y) 5 +x 5 1+ 5+1 x 5 +x 5
2 2 √ − (1 − x) ln − 5 1−x−y3 ln
1 2 1 √ 2 2 (1−y) 5 1− 5−1 x 5 +x 5 2 (1−y) 5 − 5−1 x(1−y) 5 +x 5 2 √ √ √ 1 √ 1 p √ 10+2 5x 5 10+2 5x 5
− (1 − x) arctan +2 10 − 2 5 1−x−y3 arctan √ √ 1 1 1 5 −( 5−1)x 5 (1−y) 5 4(1−y)√ 4−(√ 5−1)x 5 √ 1 √ 1
p √ 1−x−y 5 10−2 5x 10−2 5x 5 −2 10 + 2 5 √ √ 1 − (1 − x) arctan 3 arctan 1 1 (1−y) 5 4(1−y) 5 +( 5+1)x 5 4+( 5+1)x 5   1 1−y−3x 1−3x y (1−x−y)2 − (1−x)2

3

2

8

x ) arctan( 1−y

[(1 − y)− 3 ln  1−x−y 1

y (1−y) 3

√ −2 3

ln

1−x−y

1 (1−y) 3

(1−y) 8 −x 8 1−x 8 √ 1 1 1 (1−y) 4 + 2(x(1−y)) 8 +x 4 ln √ 1 1 1 (1−y) 4 − 2(x(1−y)) 8 +x 4 1 8

1−x−y 1 (1−y) 3

1 −x 3

1−x−y 1

1

− arctan x


3 − ln

1 2(1−y) 3

1 +x 3




1

1

− ln 1+√2x 81 +x 14 1− 2x 8 +x 4




√ √ 2 3 − 2 (1 − y)− 8 arctan

1−x

1 (1−x 3 )3


3 − (1 − x) ln

(1−y) 3 −x 3 √ 1 3x 3

arctan

1 8

√

4−( 5−1)x 5

x(1−y)

1 (1−y) 4

√ 1 − 2 3 (1 − y)− 3 arctan

1−x 1

(1−x 3 )3

− (1 − x) arctan

√

1

3x 3

1 2+x 3




√


18

1 −x 4

− arctan

1 +x 3

1

2x 8

1 1−x 4

1

3x 3

1 2(1−y) 3

√

− arctan




√ 1 3x 3 1 2+x 3


]

13 σ α1 α2 β1 β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y) 1 1 1  1 1 1 1
 4 4 3 x ) 4 − arctan x 4 (1 − y)− 4 ln (1−y) 1 +x 1 − ln 1+x 14 − 2 (1 − y)− 4 arctan( 1−y 2 34 1 74 2 3 4x 4 y

2

3 4

1

11 4

2

2

4 5

1

9 5

2

-

(1−y) 4 −x 4

21

7 16x 4

1 4

5x 5 y

y

 1−x−y 1 (1−y) 4

ln

1 (1−y) 4 1 (1−y) 4

 1 (1 − y)− 5 ln

1 +x 4 1 −x 4

1−x 4

1

1−x−y

− (1 − x) ln 1+x 14 − 2

1 (1−y) 4

1−x 4

1−x−y 1

1

(1−y) 5 −x 5

1

1

x ) 4 − (1 − x) arctan x 4 arctan( 1−y

1−x


5 − ln




1

(1−x 5 )5

√ √ 1 1 2 2 2 √ 1 (x(1−y)) 5 +x 5 1− 5−1 x 5 +x 5
(1−y) 5 − 5−1 2 2 √ √ − ln − 5 (1 − y)− 5 ln 2 2 2 1 1 5+1 5+1 5 +x 5 (1−y) 5 + 2 (x(1−y)) √ 1+ 2 x 5 +x 5 √ √ 1 √ 1 p √ 10+2 5x 5 10+2 5x 5
− 51 −2 10 + 2 5 (1 − y) arctan √ √ 1 1 1 − arctan 5 −( 5−1)x 5 5 4(1−y)√ 4−( √ 5−1)x √ 1 √ 1
 p √ 1 10−2 5x 5 10−2 5x 5 −2 10 − 2 5 (1 − y)− 5 arctan √ √ 1 1 − arctan 1

4(1−y) 5 +( 5+1)x 5

√ 2 1 5−1 (x(1−y)) 5 +x 5 2 √ 2 2 1 (1−y) 5 + 5+1 (x(1−y)) 5 +x 5 2

4+( 5+1)x 5

2

2

4 5

1

14 5

2

9 9 25x 5

y

5 6

1

11 6

2

1−x−y

1 (1−y) 5

1−x−y

ln

(1−y) 5 −

1

(1−y) 5

1−x−y

ln

1

1

(1−y) 5 −x 5

12x 6 y

(1−y) 3 − x(1−y) 6 +x 3
1 √ 3 x(1−y) 6

√ 1 −2 3 (1 − y)− 6 arctan 2

5 6

1

17 6

2

-

 1−x−y

55 36x

11 6

1

y (1−y) 6

√ − 3

1−x−y 1 (1−y) 6

r

ln r

2

1

15 8

2

7 7 8x 8

y



1

1

(1−y) 3 − x(1−y)
1 √ 3 x(1−y) 6

arctan

1 (1−y) 3

1

- 21 2

2 1

1

1 2

2

2 1

1

1

2

2 1

1

3 2

2

2 1

1

2

2

2 1

1

5 2

2

2 1

1

3

2

2 1

1

7 2

2

2 1

1

4

2

2 1

1

5

2

1

2

1

- 21 1 2

1

1

2

1

2

2

2 2 2 2

3 2 3 2 3 2 3 2

2

1 1 (1−y) 8

ln

1

(1−y) 8 +x 8 1 (1−y) 8

1 −x 3

1 −x 8


61


16

1

1

1




2 +x 5

1

1

− (1 − x) arctan

1+x 8

+

1 1−x 8

√1 2

1


q 1 1 1 1+x 6 +x 3 (1+x 6 )

1 (1−y) 8

(1−x−y) 2

1 y [(1

−1

1

1

1

1−x 3

ln





1 1 √ 1 (1−y) 4 + 2 x(1−y) 8 +x 4
1 1 √ 1 (1−y) 4 − 2 x(1−y) 8 +x 4
1 √ 2 x(1−y) 8 1 (1−y) 4

1 −x 4

− x − y) − (1 − x) ] √ x √  arcsin( 1−y ) arcsin( x)  √1 √ √ − xy 1−x−y 1−x (1−x)(1−y)
1 ln xy 1−x−y q √  √ √ 3 x 1 − x arcsin x − 1 − x − y arcsin 1−y 3 2 x y 
 1−x−y 2 − (1 − x) ln(1 − x) x2 y (1 − x − y) ln 1−y 3 √ x   (1−x−y) 23 arcsin √ 1−y −(1−x) 2 arcsin x 5 √ +y 2 x y x   1−x−y 3 2 2 x3 y xy − (1 − x − y) ln( 1−y ) + (1 − x) ln(1 − x)

  1−x−y 2 3 2 2 3 x4 y 2 (1 − x − y) ln( 1−y ) − (1 − x) ln(1 − x) + 2x (1 − y) − 1 + 5x y  2 2 2 1 (1−y) −6x(1−y)−3x − 1−6x−3x y (1−x−y)3 (1−x)3   1−y+x 1 1+x y (1−x−y)2 − (1−x)2 1 1 − 23 2 y [(1 − y) (1 − x − y)  √  2 √ 1−y 1 − √1−x xy 1−x−y

3

− (1 − x)− 2 ]

√

1

1

2x 8

+x 4

− ln 1+√2x 18 +x 14

− arctan

(1−x) 2

−1

1

1−x 6 +x 3 (1−x 6 )

√ 1 3x 6

− 81

(1 − y)

√ − 2








− (1 − x) ln q 1

+x 3 (1−y) 6 −x 6

1

1

1 3x 6 1 1−x 3

+x 3 (1−y) 6 +x 6

1

− ln

√

x 1 arctan( 1−y ) 8 − arctan x 8 1 arctan (1−y) 8 √ √
 x  arcsin 3 1−y 1−4x 1 1−y−4x 2 − arcsin 5x 5 y (1−x−y)2 − (1−x)2 − 3x (1−x−y) 2 (1−x) 2 √  arcsin √ x √  arcsin x x 1−y 1 − 3 3 (1−x−y)(1−x) + y

−2

2 1

2

+x 5

(1−x 6 +x 3 )(1−x 6 )

(1−y) 6 −x 6

− arctan

1

1

(1−y) 3 −x 3

(1−y) 3 + x(1−y)

1

7 8

− (1 − x) ln
1−x

√

5−1 1 x5 2 √ 1 5 1+ 5+1 x 2

1−


5 − (1 − x) ln 1 5 )5 √ (1−x √ √ 1 √ 1 p √ 1−x−y 10+2 5x 5 10+2 5x 5
+2 10 + 2 5 √ √ 1 − (1 − x) arctan 1 1 arctan 1 5 −( 5−1)x 5 (1−y) 5 4(1−y)√ 4−(√ 5−1)x 5 √ 1 √ 1
 p √ 1−x−y 10−2 5x 5 10−2 5x 5 +2 10 − 2 5 √ √ 1 arctan 1 1 − (1 − x) arctan 1 (1−y) 5 4(1−y) 5 +( 5+1)x 5 4+( 5+1)x 5


1 1 1 2 1 1 1 1 1  (1−y) 6 +x 6 (1−y) 3 + x(1−y) 6 +x 3 1 (1+x 6 +x 3 )(1+x 6 )2 5 (1 − y)− 6 ln

− ln
1 1 5 1 1 1 1 1 1 2 2

−

2

√ 5

1−

√

1

2x 8

1 1−x 4







14 σ

α1

α2

β1

β2

2

3 2

1

5 2

2

2

3 2

1

3

2

2

3 2

1

7 2

2

2

3 2

1

4

2

2

2

1

- 21

2

2

2

1

1 2

2

2 2

2 2

1 1

1 3 2

2 2

2

2

1

5 2

2

2

2

1

3

2

2

2

1

7 2

2

2

2

1

4

2

2

5 2

1

- 21

2

2

1

1 2

2

1

1

2

2

5 2 5 2 5 2

1

3 2

2

2

5 2

1

2

2

2

5 2

1

3

2

2

5 2

1

7 2

2

2

5 2

1

4

2

2

3

1

- 21

2

2

F2 (σ; α1 , α2 ; β1 , β2 ; x, y) q √ √  3 x − tanh−1 x 1 − y tanh−1 1−y 3 x2 y p √  4 1 − x − (1 − y)(1 − x − y) − y x2 y √ x √ −1 √   1−y(1−x−y) tanh−1 x 1−y −(1−x) tanh 15 √ - 2x2 y y + x
  3 3
3 1 2 2 (1 − x − y) 2 − (1 − y) 2 − 8 (1 − x) 2 − 1 − 12xy x3 y 8(1 − y) √ x √
  2 2 3 (1−y) arcsin 1−y 2−14x−3x2 1 2(1−y) −14x(1−y)−3x 2 − − 15x − arcsin 7x 7 2y (1−x−y)3 (1−x)3 (1−x−y) 2 (1−x) 2 √ x √  √ (1−y) arcsin 1−y arcsin x
 1 2(1−y)+x 2+x − 5 5 2y (1−x−y)2 − (1−x)2 + 3 x (1−x−y) 2

1 y [(1

 1

−2

− y)(1 − x − y)

3 3

 (1−y) arcsin

2x 2 y  2 x2 y ln(1

√

− (1 − x)

] √

x 1−y

(1−y) arcsin

+ √1xy √ x

1 2 (1−x)(1−x−y)

3 (1−x−y) 2

1−y

−

1−x−y

(1−x) 2

−2

−

arcsin

√
 x 3

(1−x) 2

√

arcsin x √ 1−x

1−x−y √1−yx √ √ √  − 1−x arcsin x  (1−y) 1−x−y arcsin 15 √ 1−y - 2x2 y y + x 
1−x−y 6 (1 − y)(1 − x − y) ln − (1 − x) ln(1 − x3 y 1−y   3 2 2 3 2 1 (1−y) −9x(1−y) −9x (1−y)+x +x3 − 1−9x−9x y (1−x−y)4 (1−x)4  2 2 2 1 3(1−y) +6x(1−y)−x − 3+6x−x 3y (1−x−y)3 (1−x)3 3 5 1 − 52 2 − (1 − x)− 2 ] y [(1 − y) (1 − x − y)   3−x 1 3(1−y)−x 3y (1−x−y)2 − (1−x)2




− x) − (1 − y) ln

x) − xy



3

 (1−y) 2 2 3xy (1−x−y) 32

−

1

3

(1−x) 2

3

2 2

3 3

1

1 2

 (1−y) 2 8 √ 3x2 y 1−x−y

 8(1−y)2 +9x(1−y)−2x2

2

1 y [(1

 1 5(1−y)−2x 8y (1−x−y)2

2

2

3

1

3 2

2

3

1

2

2

2

3

1

5 2

2

2

3

1

7 2

2

2

3

1

4

2

2

7 2

1

- 21

2

2

1

1 2

2

2

7 2 7 2

1

1

2

2

7 2

1

3 2

2

2

7 2

1

2

2

2

7 2

1

5 2

2

+y



 3 2 2 3 1 8(1−y) −80x(1−y) −39x (1−y)+6x 8y (1−x−y)4 1 8y

1

√1 1−x

√ x  (1−y) 32 tanh−1 √ 1−y  −tanh−1 x 5 √ +y x2 y x  1 3 8 2 2 2 x3 y 2 (1 − y) − (1 − x − y) (1 − y)

2

1

−



−

(1−x−y)3

8+9x−2x2 (1−x)3

−


 1 + (1 − x) 2 − 1 + xy

8−80x−39x2 +6x3 (1−x)4

√ + 15 x

− y)2 (1 − x − y)−3 − (1 − x)−3 ]

 1 2(1−y)−x 2y (1−x−y)2

−

−

5−2x (1−x)2

2−x (1−x)2



√

 1 (1−y)2 arcsin 3 √ 3 8xy x (1−x−y) 2

+

x 1−y

√3 x

−

(1−y)2 arcsin

√

(1−y) arcsin

x 1−y

5 (1−x−y) 2

arcsin

√
x

3 (1−x) 2

+

2

3

− 105x 2 √ x

1−y 1−x−y

1−y

7

(1−x−y) 2

− −

arcsin

√ 5

(1−x) 2

1 1−x



(1−y)2 arcsin

9 (1−x−y) 2

−

√ arcsin x


5

−

−

1

5 (1−x) 2

5−3x (1−x)2





7

(1−x) 2

x


√
x  1 (1−y)2 arcsin √ 1−y  x 15 √ √ √ +y − arcsin 8x2 y x 1−x−y 1−x 
 1−x−y 3 2 x3 y xy + ln(1 − x) − (1 − y) ln 1−y   4 3 2 2 3 4 2 +20x3 −3x4 1 5(1−y) −60x(1−y) −90x (1−y) +20x (1−y)−3x − 5−60x−90x 5y (1−x−y)5 (1−x)5   3 2 2 3 2 1 5(1−y) −15x(1−y) −5x (1−y)+x +x3 − 5+15x−5x 5y (1−x−y)4 (1−x)4 5 7 1 − 72 2 − (1 − x)− 2 ] y [(1 − y) (1 − x − y)  15(1−y)2 −10x(1−y)+3x2 2 1 − 15−10x+3x 15y (1−x−y)3 (1−x)3

 (1−y) 2 2 5xy (1−x−y) 52  1 5(1−y)−3x 5y (1−x−y)2

√

x 1−y

−

arcsin

√
 x 9

(1−x) 2

15 σ α1 α2 β1 β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)  (1−y) 25  8 1 2 72 1 3 2 3 − 3 + y 2 15x y (1−x−y) 2

 2(1−y) 52

2

7 2

1

4

2

8 √ 5x3 y 1−x−y

2

7 2

1

9 2

2

7 3x3 y

2

4

1

- 21 2

 (1−y) 52 3

−

2

4 4

1

1 2

√

2

1 y [(1

2

2

4

1

3 2

2

4

1

2

2

2

4

1

5 2

2

2

4

1

3

2

2

4

1

7 2

2

2

4

1

9 2

2

9 4

3 4

1

1 2

2

9 4

3 4

1

3 2

2

5 2

- 21 1

1 2

2

5 2

- 21 1

3 2

2

5 2

- 21 1

5 2

2

5 2

- 21 1

7 2

2

5 2

1 2

1

- 12 2

5 2

1 2

1

3 2

2

5 2

1 2

1

5 2

2

5 2

1 2

1

7 2

5 2

1

1

- 12 2

5 2

1

1

1 2

2

5 2 5 2

1

1

3 2

2

1

1

2

2

5 2

1

1

5 2

2

5 2

1

1

3

2

5 2

1

1

7 2

2

5 2

1

1

4

2

5 2

3 2

1

- 12 2

2

−1 x 1−y −tanh

√

(1−x−y)

−

11 2

(1−x)

 48(1−y)3 +87x(1−y)2 −38x2 (1−y)+8x3

1 48y

1

√


 − 2 (1 − y)2 − 1 + xy

x − 3 (1 − y)2 − 1 − xy x  16(1−y)4 −208x(1−y)3 −165x2 (1−y)2 +50x3 (1−y)−8x4 2 +50x3 −8x4 1 − 16−208x−165x 16y (1−x−y)5 (1−x)5 √ 3 √ x 3 (1−y) arcsin arcsin x
 1−y 2

2

1

√2 1−x

tanh−1

−315x

2

(1−x) 2

(1−x−y)4

 (1−y)3 1 3xy (1−x−y)3

−

1 (1−x)3



√

x  3 (1−y)3 arcsin 1−y 1 √ 5 16xy x (1−x−y) 2   1 3(1−y)−2x 3−2x 3y (1−x−y)2 − (1−x)2

5 √3 48x2 y x



−

(1−y)3 arcsin

√

3 (1−x−y) 2

−

48+87x−38x2 +8x3 (1−x)4

−

x 1−y

√

33−26x+8x2 (1−x)3

√

arcsin



11 2

− y)3 (1 − x − y)−4 − (1 − x)−4 ]

 33(1−y)2 −26x(1−y)+8x2 1 48y (1−x−y)3




x


5 (1−x) 2

+

−

15 √ x

√ + 105 x

(1−y)3 arcsin

√

3(1−y)2 −14x(1−y)+8x2 (1−x−y)2

arcsin



√1 (1+ 1−x)2

− (3 − 2x) − 3



(1−x−y) tanh−1

 3 3
8 (1 − x − y) 2 − (1 − x) 2 −  2 2 2 2 (1−y) −8x(1−y)−8x − 1−8x−8x 7 7 3y 4 3x3 y

(1−x−y) 2

3−14x+8x2 (1−x)2



3

2 − 12 (1 − x − y)−1 − (1 − x)−1 ] 3y [(1 − y)  1 1 4 − 21 − (1 − x)− 2 − (1 − y)− 2 3xy (1 − x − y) √ √ −1 −1 x   x 1−y −tanh 2 tanh 1 √ − √1−y +1 xy x

 3(1−y)−2x 5 √ 3x2 y 1−y

x


7

(1−x) 2

(1−x) 2 √ √ √ 1+ x− 1− x  √ 1−x




(1−x) 2

+1

√

 x

3(1−y)−4x arcsin

 2 2 −6x(1−y)−3x2 2 (1−y) √ − 1−6x−3x 3y (1−x)3 1−y(1−x−y)3   2 √ 1−y+x 1+x − (1−x) 2 3y 1−y(1−x−y)2

−

√

√

1 1 2 −1 (1 − x − y)− 2 − (1 − x)− 2 ] 3y [(1 − y) √ √ √ x   √ x 1−y −arcsin 1 arcsin √ + 1−x − 1−x−y xy 1−y x
 3(1−y)−2x √1−x−y √ 5 − (3 − 2x) 1 − x − 8x2 y 1−y



arcsin

−

√
x

1−2x 2 √ 1−y−2x −√ 3y 1−x−y(1−y)2 1−x 1 1 2 −2 (1 − x − y) 2 − (1 − x) 2 ] 3y [(1 − y) √ √ x  arcsin √ 1−y √ −arcsin x 1−x−y 1 √ − (1−y−2x) − (1 − 2x) 1 − 4xy (1−y)2 x
 3(1−y)2 −4x(1−y)+4x2 √1−x−y √ 5 − (3 − 4x + x2 ) 1 − x) 48x2 y (1−y) √ 2x √ x
 (1−y−2x) arcsin 1−y −(1−2x) arcsin √ −3 x  1−y−4x  2 1−4x 3y (1−x−y) 52 − (1−x) 52

1 √ 8 √ √ 3y 1−y( 1−y+ 1−x−y)2

+

x 1−y

9 (1−x−y) 2

−

7 (1−x−y) 2

√

2 2 2 − 3(1−y) +2x(1−y)−8x + 3+2x−8x 3 1−x−y 1−x (1−x) 2 √

 x √ − arcsin − 3x (1 − y)2 − 1 + 2x2 y 1−x √ 3 √ 3 (1− x) 2 +(1+ x) 2 

−

x  √ (1−y)3 arcsin 1−y 35 √ 3 x 4 48x y 1−x−y √ √ 3 √ 3  √ 2 ( 1−y− x) 2 +( 1−y+ x) 2 − 3√ 5y (1−x−y) 2 1−y √ √  √1−y+√x− √1−y−√x √4 √ − 5 xy (1−y)(1−x−y)



x 1−y

(1−y)3 arcsin

x

√1−y x

−(3−4x) arcsin



−1 x 1−y −(1−x) tanh

√

√

x 8(1−y)2 −12x(1−y)+3x2 √ 1−y

√

x

+ 8 − 12x + 3x2



√ x

arcsin

√
 x 9

(1−x) 2

16 σ

α1

α2

β1

β2

5 2

3 2

1

1 2

2

5 2

3 2

1

3 2

2

5 2

3 2

1

5 2

2

5 2

3 2

1

7 2

2

5 2

2

1

- 12

2

5 2 5 2

2

1

1 2

2

2

1

1

2 2

5 2

2

1

3 2

5 2

2

1

5 2

2

5 2

2

1

3

2

5 2

2

1

7 2

2

5 2

2

1

4

2

5 2

5 2

1

- 21

2

5 2

5 2

1

1 2

2

5 2

5 2

1

3 2

2

5 2

5 2

1

7 2

2

5 2

3

1

- 12

2

5 2

3

1

1 2

2

5 2

3

1

1

2

3

1

3 2

2

3

1

2

2 2

5 2 5 2 5 2

3

1

5 2

5 2

3

1

7 2

2

5 2

3

1

4

2

5 2

7 2

1

- 21

2

5 2

7 2

1

1 2

2

5 2 5 2

7 2 7 2

1

2

1

3 2 5 2

5 2

4

1

- 12

2

5 2

4

1

1 2

2

5 2

4

1

1

2

5 2

4

1

3 2

2

5 2

4

1

2

2

5 2

4

1

5 2

2

2

F2 (σ; α1 , α2 ; β1 , β2 ; x, y)  1−y+2x  1+2x 2 5 − 5 3y (1−x−y) 2 (1−x) 2 2 − 23 − (1 [(1 − x − y) 3y

3

− x)− 2 ]

√ x √  arcsin x 1 1 1−y −arcsin √ (1 − x − y)− 2 − (1 − x)− 2 − x
√ x   3(1−y)−2x arcsin √ 1−y √ √ −(3−2x) arcsin x 5 √ − 3( 1 − x − y − 1 − x) 2x2 y x
 √1−y (1−y)2 −10x(1−y)−15x2 2 2 − 1−10x−15x 3y (1−x−y)4 (1−x)4   √1−y(1−y+3x) 2 1+3x − (1−x) 3 3y (1−x−y)3   1 2(1−y)+x 2+x 5 − 5 3y 2 xy

(1−x−y) 2 (1−x) 2 1 2 −2 2 − (1 − x)−2 ] 3y [(1 − y) (1 − x − y) √ x −1 √   √1−y x tanh−1 1−y −tanh 1 1 √ − − xy 1−x−y 1−x x  2(1−y)−x  √ 8 √ − 2( 1 − y − 1) − √2−x 3x2 y 1−x−y 1−x
√ x  3(1−y)−x tanh−1 √ 1−y √ −(3−x) tanh−1 x 5 √ − 3( 1 2 2x y x   √ √ 16 √ 1 − x − y)( − 3) − (1 + 1 − x) 3y ( 1 − y + 2 2  2 (1−y) (1−y) −12x(1−y)−24x 9 3y (1−x−y) 2   2 (1−y)(1−y+4x) − 1+4x7 7 3y 2 2 (1−x−y) (1−x)




−

1−12x−24x2 9

(1−x) 2

5

2 3y [(1

 − y − 1)



5

− y)(1 − x − y)− 2 − (1 − x)− 2 ] √ x √  3(1−y)−x x (1−y) arcsin 1−y −arcsin 5 3−x √ √ √ − 3 − 3x2 y 1−x−y x 1−x
3 2 2  2 2 (1−y) 2 (1−y) −14x(1−y)−35x − 1−14x−35x 3y (1−x−y)5 (1−x)5 3   2 (1−y) 2 (1−y−5x) 1+5x − (1−x) 4 3y (1−x−y)4  8(1−y)2 +8x(1−y)−x2 2 1 − 8+8x−x7 7 12y (1−x−y) 2 3 2 −3 2 3y [(1 − y) (1 − x − y)   4−x 1 4(1−y)−x 6y (1−x−y) 52 − (1−x) 25 1 4xy

 √1−y(1−y+x)

(1−x) 2

− (1 − x)−3 ]

1+x (1−x)2

− (1−x−y)2
√  1−y 3(1−y)−x 5 8x2 y 1−x−y

−

 8(1−y)2 −4x(1−y)−x2 4 √ 3x3 y 1−x−y

−

3−x 1−x

−

tanh−1

2 2 − 72 3y [(1 − y) (1 − x − y)  5(1−y)−2x  2 5−2x 15y (1−x−y) 52 − (1−x) 25

−1 x 1−y −tanh

√ x

2 8−4x−x √ 1−x




−

5

3

− 8 (1 − y) 2 − 1  1−16x−48x2 (1−x)

−(3+x) tanh−1




√ x

11 2

7

5




−

  2 (1−y) 2 (1−y+7x) 1+7x − (1−x) 5 3y (1−x−y)5  16(1−y)3 +24x(1−y)2 −6x2 (1−y)+x3 1 9 24y (1−x−y) 2

1−18x−63x2 (1−x)6

−



16+24x−6x2 +x3 9 (1−x) 2

5 2 −4 2 − (1 − x)−4 ] 3y [(1 − y) (1 − x − y)  8(1−y)2 −4x(1−y)+x2 2 1 − 8−4x+x7 7 12y 2 2 (1−x−y) (1−x)

 √1−y

x

√1−y x

− (1 − x)− 2 ]

2 2  2 (1−y) 2 (1−y) −18x(1−y)−63x 3y (1−x−y)6

1 24xy

√ x

√




3(1−y)+x tanh−1

−

2 2 2  2 (1−y) (1−y) −16x(1−y)−48x 11 3y (1−x−y) 2   2 2 (1−y) (1−y+6x) − 1+6x9 9 3y (1−x−y) 2 (1−x) 2

√

3(1−y)2 +8x(1−y)−3x2 (1−x−y)3




−

3+8x−3x2 (1−x)3

−3



tanh−1

√

−1 x 1−y −tanh

√ x

√ x


17 σ

α1

α2

β1

β2

5 2

4

1

3

2 2

5 2

4

1

7 2

3

- 21

1

1 2

2

3

- 21

1

1

2

F2 (σ; α1 , α2 ; β1 , β2 ; x, y)   2−x 1 2(1−y)−x 5 − 5 3y

(1−x−y) 2 (1−x) 2
√ 2  1−y 3(1−y) −2x(1−y)+3x2 5 48x2 y (1−x−y)2




 2(1−y)−3x 1 4y (1−y)2 (1−x−y)

−

 2(1−y)−3x 1 4y (1−y) 52 √1−x−y

−

√

1−x


1 −2 −1 + 8y 3 (1 − y)

(1−y−3x) tanh−1

2

- 21

1

2

2

3

- 21

1

5 2

2

(1−y+3x)(1−y−x) tanh 3 √1 5 32xy x (1−y) 2

3

- 21

1

3

2

2 15x2 y

3

- 21

1

7 2

2

3

√1 x

5

1 2y [(1

1

1

4

2

3

1

- 21

2

3

1 2 1 2

1

1

2

3

1 2

1

3 2

2

1−y−5x 1−5x 1 2y (1−x−y)3 − (1−x)3  2(1−y)−x 1 2−x
32 − 3 4y 2 (1−x) (1−y)(1−x−y)

3

1 2

1

2

2

3

1 2

1

5 2

2

3

1 2

1

3

2

3

1 2

1

7 2

2

3

1 2

1

4

2

3

1

1

- 21

2

3

1

1

1 2

2

3

1

1

3 2

2

1

1

2

2

3

1

1

5 2

2

3

1

1

3

2

3

1

1

7 2

2

3

1

1

4

2

3

1

- 21

2

1

1 2

2

3

3 2 3 2 3 2

1

1

2

3

3 2

1

2

2

3

3 2

1

5 2

2

3

5 (1−y) 2

1 2y [(1

3

1 2y [(1

−1

+ 3x2 )

− (1 − x)2 (1 − x) tanh−1

− (4 + 3x)(1 − x) 2 + 4y



−

−1 x 1−y −(1+x) tanh

√

x

1 1−x

+



√

x 1−y 3 (1−y) 2

tanh−1

√1 x

1

√

x


√
 x

√
x + 1 − 3x −

(1−y−3x)  (1−y)2



5

3



3

5




4(1−y)+3x (1−y−x) 2

1 1 4y (1−y)(1−x−y)

√

− (1 − 3x) tanh−1

− (1 + 3x)(1 − x) tanh−1

5 (1−y) 2

4 35x3 y

3

1−y

(2(1−y)+3x)(1−x−y) 2

3

(2 + 3x)(1 − x) 2 −

- 21



x 1−y

1

 3(1−y)2 −2x(1−y)+3x2 5 − (3 − 2x 64x2 y (1−y)2 √ x 2 −1 (1−x−y) (1−y+x) tanh 1−y − √3x 5 (1−y) 2 

√

5 (1−y) 2

− y)− 2 (1 − x − y) 2 − (1 − x) 2 ] √ x −1  

−3

√ x √ tanh−1 1−y √  − 3 x( − tanh−1 x) 5 (1−y) 2  2−3x

1

3

−

(1+x−y) tanh−1

2−3x 1−x

3 2

- 21

3−2x+3x2 (1−x)2

√
 x



− tanh−1

1

√
 x

− y)− 2 (1 − x − y)− 2 − (1 − x)− 2 ] x  1 (1−y+x) tanh−1 √ 1−y  √
y 3 √ − (1 − x) tanh−1 x − 1−y 3 8xy x (1−y) 2
 2(1−y)+x √1−x−y  √ − (2 + x) 1 − x - 3x22 y 3 (1−y) 2
√ x  √
 (1−x−y) 3(1−y)+x tanh−1 3(1−y)−x 1−y 1 15 √ − (1 − x)(3 + x) tanh−1 x − (3 − x) − 3 2 32x y 1−y x (1−y) 2  
1−x−y
32 3 4 − (4 + x)(1 − x) 2 + 4y 5x3 y 4(1 − y) + x 1−y √ x √
  2 2 3 arcsin 1−y 2−14x−3x2 1 2(1−y) −14x(1−y)−3x 2 − (1−x)3 − 15x − arcsin 7x 7 4y (1−y)(1−x−y)3 2 2 √ x (1−x−y)√
 (1−x)  √ arcsin 2(1−y)+x x arcsin 1−y 1 2+x 4y (1−y)(1−x−y)2 − (1−x)2 + 3 x (1−x−y) 25 − (1−x) 52 √ x √  arcsin arcsin x
 1−y 1 1 1 √1 − − + 3 3 4y (1−y)(1−x−y) 1−x x (1−x−y) 2

− y)

−1

(1 − x − y)

−1

− (1 − x)

(1−x) 2

]

√ x √
 arcsin x 1−y y 3 √1 √ √ − 1−y − arcsin 4xy x 1−x−y 1−x  xy
 1−x−y + ln (1−x)(1−y) - x12 y 1−y √ x √ √ √  3(1−y)−x 1−x−y arcsin x 1−y − 1−x arcsin 5 √ − (3 − x) − 3 2 4x y 1−y x 

1−x−y 3 − (1 − x) ln(1 2x3 y 2 (1 − x − y) ln 1−y  2 2 2 1 (1−y) −10x(1−y)−15x − 1−10x−15x 2y (1−x−y)4 (1−x)4   1−y+3x 1+3x 1 2y (1−x−y)3 − (1−x)3  2(1−y)+x  2+x 1 4y √1−y(1−x−y) 52 − (1−x) 25



1 2y [(1




1

3

3


− x) −

− y)− 2 (1 − x − y)− 2 − (1 − x)− 2 ] √ x  √
 tanh−1 y 1−y 3 √1 √ − − tanh−1 x 4xy (1−x−y)(1−x) x 1−y

x2 y 1−y



18 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)  2(1−y)−x  2 √ − √2−x x2 y 1−x (1−y)(1−x−y)
√ x −1 

σ

α1

α2

β1

β2

3

3 2

1

3

2

3

3 2

1

7 2

2

3

3 2

1

4

2

3

2

1

- 21

2

 2 2 1 4(1−y) −48x(1−y)−61x 8y (1−x−y)4

3

2

1

1 2

2

3

2

1

1

2

 1 4(1−y)+11x 8y (1−x−y)3

3

2

1

3 2

3

2

1

3

2

3

2

3(1−y)−x tanh √ 1−y

15 5

8x 2 y

- x43 y


q 1−x−y 4(1 − y) − x 1−y



−

1+x (1−x)3

2

1 3 8y (1−x−y)2

−

3 (1−x)2

2

2

1 2y [(1

1

5 2

2

y 3 8xy (1−x−y)(1−x)

1

3

2

1 x(1−x)(1−x−y)

2

3

2

1

7 2

3

2

1

4

2

3

5 2

1

- 21

2

3

5 2

1

1 2

2

3

5 2

1

1

2

3

1

3 2

2

3

5 2 5 2

1

2

2

3

5 2

1

3

2

3

5 2

1

7 2

2

3

5 2

1

4

2

3

5 2

1

5

2

3

3

1

- 21

2



5 8x 2

3 x3 y

y



4−48x−61x2 (1−x)4



+



+

(1−y+2x) arcsin √1 5 x (1−x−y) 2

√1 x

(1−y−2x) arcsin

1 x2 y

ln

1−x−y (1−x)(1−y)




√

√

x 1−y

3 (1−x−y) 2

√

x 1−y

−




(3−2x) arcsin √ 1−x

x 1−y

−

√

x 1−y

 3 2 2 3 1 (1−y) −15x(1−y) −45x (1−y)−5x 2y (1−x−y)5  2 2 2 1 (1−y) +6x(1−y)+x − 1+6x+x 2y (1−x−y)4 (1−x)4

−

−

5 (1−x) 2




1−15x−45x2 −5x3 (1−x)5

3

1

1 2

2

3

3

1

1

2

1

3 2

2

3

3

3

3

1

2

2 2

2

3

3

1

5 2

3

3

1

7 2

√

  1 (1−y) 2 2(1−y)+3x) − 2+3x7 7 4y (1−x−y) 2 (1−x) 2   1 3(1−y)+x 3+x 6y (1−x−y)3 − (1−x)3 5 1 1 − 52 2 − (1 − x)− 2 ] 2y [(1 − y) (1 − x − y)
√  2−3x 1−y 2(1−y)−3x  2 − 3 3 2 3x y (1−x) 2 (1−x−y) 2 √ x √ √  1−y tanh−1 −tanh−1 x  xy 5 √1−y − 3 2 4x y (1−x−y)(1−x) x
3 
3  4 √ 1 1 1 √ √ √ − 1−x 1+√11−x y 1−x−y 1−y+ 1−x−y 
3 
4 1 8 √ √ − 1+√11−x y 1−y+ 1−x−y  16(1−y)3 −272x(1−y)2 −659x2 (1−y)−30x3 2 1 −30x3 − 16−272x−659x 32y (1−x−y)5
(1−x)5 √ x √ 3 (1−y) 5(1−y)+4x arcsin (5+4x) arcsin x
 1−y −105x 2 − 11 11 (1−x−y) 2 (1−x) 2  2 2 2 16(1−y) +83x(1−y)+6x 1 − 16+83x+6x 32y (1−x−y)4 (1−x)4
√ x √
 √ (1−y) 3(1−y)+4x arcsin 1−y x − (3+4x) arcsin +15 x 9 9 (1−x−y) 2 (1−x) 2   1+2x 1 (1−y)(1−y+2x) − (1−x) 4 2y (1−x−y)4  13(1−y)+2x 1 13+2x 32y (1−x−y)3 − (1−x)3 √ x √
 (1−y)(1−y+4x) arcsin x 1−y + √3x − (1+4x) arcsin 7 7 (1−x−y) 2 (1−x) 2 1 −3 −3 − (1 − x) ] 2y [(1 − y)(1 − x − y)  1−y+2x 3 1+2x 32xy (1−x−y)2 − (1−x)2 √ x √
 (1−y)(1−y−4x) arcsin x 1−y − √1x − (1−4x) arcsin 5 5 (1−x−y) 2 (1−x) 2  3(1−y)−2x 3−2x 15 − 1−x 32x2 y 1−x−y √ x √
 (1−y) 3(1−y)−4x) arcsin x 1−y − √1x − (3−4x) arcsin 3 3 (1−x−y) 2 (1−x) 2

√
 x

x




1

3

7 (1−x) 2

(1+2x) arcsin

3 (1−x) 2

x 1−y

x 1−y

(3+2x) arcsin

(1−2x) arcsin

x



(2 − x) ln(1 − x) − 2(1 − y) − x ln 1 −

−

√

9 (1−x−y) 2

7 (1−x−y) 2

−

3(1−y)−2x arcsin √ 1−x−y




5(1−y)+2x arcsin

3

− 15x 2
√

3(1−y)+2x arcsin

− x − y)−2 − (1 − x)−2 ]



15

−

√  − (3 − x) tanh−1 x  √ − (4 − x) 1 − x + 4y

√ +3 x

4+11x (1−x)3

−

1+x−y 1 2y (1−x−y)3



1−y

√

−

x


(5+2x) arcsin 9 (1−x) 2

√
 x

19 σ

α1

α2

β1

β2

3

3

1

4

2

3

7 2

1

- 21

2

3

7 2

1

1 2

2

3

7 2

1

1

2

3

1

3 2

2

1

2

2

3

7 2 7 2 7 2

1

5 2

2

3

7 2

3

7 2

1

4

2

3

4

1

- 21

2

3

F2 (σ; α1 , α2 ; β1 , β2 ; x, y)
  1−x−y x2 y 3 2x3 y (1−x)(1−x−y) + 2 (1 − y) ln 1−y − ln(1 − x)   4 3 2 2 3 4 2 −20x3 +x4 1 (1−y) −20x(1−y) −90x (1−y) −20x (1−y)+x − 1−20x−90x 2y (1−x−y)6 (1−x)6  5(1−y)3 +45x(1−y)2 +15x2 (1−y)−x3 5+45x+15x2 −x3  1 − 10y (1−x−y)5 (1−x)5
3   2 2(1−y)+5x 1 (1−y) − 2+5x9 9 4y (1−x−y) 2 (1−x) 2  15(1−y)2 +10x(1−y)−x2 15+10x−x2  1 − (1−x)4 30y (1−x−y)4 3 1 − 27 2 2y [(1 − y) (1 − x − y)   5(1−y)−x 1 5−x 10y (1−x−y)3 − (1−x)3

7

− (1 − x)− 2 ]

3

1

3

2

3

4

1

1 2

2

3

4

1

1

2

3

4

1

3 2

2

3

4

1

2

2

3

4

1

5 2

2

3

4

1

3

2

3

4

1

7 2

2

7 2

- 12

1

1 2

2

7 2

- 12

1

3 2

2

7 2

- 12

1

5 2

2

7 2

- 12

1

7 2

2

7 2

1 2

1

- 21

2

7 2

1 2

1

3 2

2

7 2

1 2

1

5 2

2

7 2

1 2

1

7 2

2

7 2

1

1

- 21

2

7 2

1

1

1 2

2

7 2

1

1

3 2

2

7 2 7 2 7 2

1

1

2

2

1

1

5 2

2

1

1

3

2

7 2

1

1

7 2

2

7 2

1

1

4

2

4 2−5x 15x2 y (1−x) 52



−

(1−y) 2 2(1−y)−5x 5 (1−x−y) 2 3









  4−5x (1−y) 2 4(1−y)−5x 8 − 4y 3 5x3 y (1−x) 32 − (1−x−y) 2  32(1−y)4 −704x(1−y)3 −2553x2 (1−y)2 −260x3 (1−y)+20x4 2 1 −260x3 +20x4 − 32−704x−2553x 64y (1−x−y) (1−x)6
6 √ 2 √ x 5(1−y)+6x arcsin 3 (1−y) (5+6x) arcsin x
 1−y − −315x 2 13 13 (1−x−y) 2 (1−x) 2  3 2 2 3 2 32(1−y) +247x(1−y) +40x (1−y)−4x 1 −4x3 − 32+247x+40x 64y (1−x−y)5 (1−x)5 √ √
 x √ (1−y)2 (1−y+2x) arcsin 1−y x − (1+2x) arcsin +105 x 11 11 (1−x−y) 2 (1−x) 2   2 1 (1−y) (1−y+3x) 1+3x − (1−x) 5 2y (1−x−y)5 √ x 2  81(1−y)2 +28x(1−y)−4x2 1−y 81+28x−4x2 15 (1−y) (1−y+6x) arcsin 1 √ − − + 9 4 4 192y (1−x−y) (1−x) x

(1+6x) arcsin 9 (1−x) 2

(1−x−y) 2

1 2y [(1

− y)2 (1 − x − y)−4 − (1 − x)−4 ]

 3(1−y)2 +16x(1−y)−4x2 1 64xy (1−x−y)3

−

3+16x−4x2 (1−x)3

 3(1−y)2 −4x(1−y)+4x2 5 64x2 y (1−x−y)2

−

3−4x+4x2 (1−x)2

 1 3(1−y)−x 6y (1−x−y)3

−

3−x (1−x)3



+

−  3−12x+8x2

(1−y)2 (1−y−6x) arcsin √3 7 x (1−x−y) 2

√3 x

(1−y)2 (1−y−2x) arcsin

−

3−2x

3 (1−x) 2 1



√

x 1−y

x 1−y

5 (1−x−y) 2

 3(1−y)2 −12x(1−y)+8x2 2 − 3 3 15y (1−y)3 (1−x−y) 2 (1−x) 2   3(1−y)−4x 2 3−4x √ √ 15y (1−y)3 1−x−y − 1−x 1 1 2 −3 (1 − x − y) 2 − (1 − x) 2 ] 5y [(1 − y) √ √ x  arcsin 1−y 3 2 −arcsin x
−10x2 (1−y)+8x3 1 √ √ − 3(1−y) −x(1−y) 24x2 y 3 x (1−y)3 1−x−y  1−y−6x  2 1−6x 5y (1−x−y) 27 − (1−x) 72  3(1−y)−2x 2 15y (1−y)2 (1−x−y) 32 2 −2 (1 − x 5y [(1 − y)

√

+

 8(1−y)2 −4x(1−y)−x2 4 3 5x3 y (1−y) 2

−

(1−6x) arcsin 7 (1−x) 2

(1−2x) arcsin 5 (1−x) 2

2 3−x−10x +8x3 √ 1−x

1

− y)− 2 − (1 − x)− 2 ]
√ √ x  arcsin √ 1−y  √ −arcsin x 3(1−y)+2x 1−x−y 1 √ − + (3 + 2x) 1 − x 4x2 y 3 (1−y)2 x   3 2 2 3 1−9x−9x2 +x3 2 (1−y) −9x(1−y) −9x (1−y)+x − 3 4 5y (1−x) (1−y) 2 (1−x−y)4  3(1−y)2 +6x(1−y)−x2 2 2 − 3+6x−x 3 3 15y (1−x) 3 2 (1−y) (1−x−y)   3(1−y)−x 3−x 2 15y (1−y) 32 (1−x−y)2 − (1−x)2   3 3 4 − 32 − (1 − x)− 2 − (1 − y)− 2 + 1 15xy (1 − x − y) 2 − 32 (1 − x − y)−1 − (1 − x)−1 ] 5y [(1 − y)   8 2 √ 2 − 2(1−y)+x +2+x − √1−x 3 15x2 y 1−x−y (1−y) 2 √ x  tanh−1 √ 1−y −tanh−1 x
2 √ − 3(1−y)+x + 3 3 2 3x y x (1−y) 2

−

 3+x  √ √ − (8 − 4x − x2 ) − 8( 1 − x − y − 1 − x)



√
 x

√
 x

√

x


20 F2 (σ; α1 , α2 ; β1 , β2 ; x, y) 
8 − 12 −1 − 15x2 y 2 (1 − y)

σ

α1

α2

β1

β2

7 2

2

1

3

2

7 2

2

1

7 2

2

7 2

2

1

4

2

7 2

2

1

5

2

7 2

5 2

1

- 21

2

7 2

5 2

1

1 2

2

7 2

5 2

1

3 2

2

7 2

5 2

1

5 2

2

7 2

5 2

1

7 2

2

− x − y)− 2 − (1 − x)− 2 ] q  3 √ 2 x √ (arcsin 3x2 y 1−y − arcsin x) − x

1

- 21

2

2 5y

2

2 5y

7 2

3

7 2

3

1

1 2

7 2

3

1

1

2

7 2 7 2

3

1

3 2

2

3

1

2

2

7 2

3

1

5 2

2

7 2

3

1

7 2

2

7 2

3

1

4

2

1

- 21

7 2

1

1 2

2

7 2

7 2

1

3 2

2

7 2

7 2

1

5 2

2

7 2

4

1

- 21

2 2 2

7 2

7 2

7 2

2

2(1−y)−3x

+ 2−3x3 2 √ x(1−x) −1

3 (1−x−y) 2

−1 √   3(1−y)−2x x tanh 1−y −tanh 1 3−2x √ √ − − 3 x2 y 1−x x 1−y(1−x−y)  4(1−y)−3x  16 4−3x √ √ −√ − 4(1−y)−x +4−x 5x3 y 1−x−y 1−y 1−x  (√1−y−√1−x−y)4  √ 32 √ − (1 − 1 − x)4 5x4 y 1−y  3 2 2 3 2 3 2 (1−y) −18x(1−y) −72x (1−y)−16x − 1−18x−72x11−16x 11 5y (1−x−y) 2 (1−x) 2

 3(1−y)2 +24x(1−y)+8x2 2 9 15y (1−x−y) 2   3(1−y)+2x 2 3+2x 15y (1−x−y) 72 − (1−x) 27

−

3+24x+8x2

5

2 5y [(1

 √1−y

9

(1−x) 2



5

3(1−y)−4x

(1−y)3 −21x(1−y)2 −105x2 (1−y)−35x3 (1−x−y)6

 √1−y

4

1

7 2

4

1

1




2

4

1

3 2

7 2

4

1

2

2

7 2 7 2

4

1

5 2

2

4

1

3

2

1

7 2

2

7 2

4

4

- 21

1

1 2

4

- 21

1

1

2

4

- 21

1

3 2

2

4

- 21

1

2

2

2

−

3−4x 3

(1−x) 2




1−21x−105x2 −35x3 (1−x)6

(1−y)2 +10x(1−y)+5x2 − (1−x)5 (1−x−y)5  8(1−y)2 +24x(1−y)+3x2  1 8+24x+3x2 − 9 9 20y (1−x−y) 2 (1−x) 2   √1−y(1−y+x) 1+x 2 − (1−x)4 5y (1−x−y)4  4(1−y)+x  1 4+x 10y (1−x−y) 27 − (1−x) 72

− y) 2 (1 − x − y)−3 − (1 − x)−3 ] √ q  3 1−y 3(1−y)−5x) −1 √ 1 x √ (tanh−1 − tanh − x) − 2 4x y 1−y (1−x−y)2 x 
 2 2 √ 2 8(1−y) −12x(1−y)+3x 4 − 8−12x+3x 3 3 5x3 y 8( 1 − y − 1) − (1−x−y) 2

2 5y



1

2 5y [(1

 (1−y)

(1−y)3 −24x(1−y)2 −144x2 (1−y)−64x3 (1−x−y)

13 2

2 2  2 (1−y) (1−y) +12x(1−y)+8x 11 5y 2 (1−x−y)




−

11 (1−x) 2




−

1−24x−144x2 −64x3 (1−x)

13 2




−



1−27x−189x2 −105x3 (1−x)7

 (1−y) 2 3(1−y)2 +42x(1−y)+35x2 2 2 − 3+42x+35x 15y (1−x−y)6 (1−x)6  16(1−y)3 +72x(1−y)2 +18x2 (1−y)−x3  2 1 −x3 − 16+72x+18x 11 11 40y (1−x−y) 2 (1−x) 2






  (1−y) 2 3(1−y)+5x 3+5x 2 − (1−x) 5 15y (1−x−y)5  24(1−y)2 +12x(1−y)−x2 2 1 − 24+12x−x 9 9 60y (1−x−y) 2 (1−x) 2 3 2 −4 2 − (1 − x)−4 ] 5y [(1 − y) (1 − x − y)   6(1−y)−x 1 6−x 15y (1−x−y) 27 − (1−x) 72 1 √3 (tanh−1 24x2 y x








3 2 2 3  2 (1−y) 2 (1−y) −27x(1−y) −189x (1−y)−105x 5y (1−x−y)7




3−5x (1−x)2

(1−x) 2

1+12x+8x2

 (1−y) 3(1−y)+4x  2 − 3+4x9 9 15y (1−x−y) 2 (1−x) 2 7 2 − 72 − (1 − x)− 2 ] 5y [(1 − y)(1 − x − y)

3

7 2




−  1+10x+5x2

3

1 2

3

(1−x−y) 2

3

7 2



q

x 1−y

−1

− tanh

√ x) −

√

1−y 3(1−y)2 −8x(1−y)−3x2 (1−x−y)3




−

3−8x−3x2 (1−x)3

√ x −1 √ √ tanh−1 1−y x  − tanh − − 15 x( ) 7 2 (1−x) (1−y) 2   2 2 2 8(1−y) −24x(1−y)+15x 1 − 8−24x+15x 7 3 3 24y (1−y) 2 (1−x−y) 2 (1−x) 2 √ x  13(1−y)−15x √
 (1−y−5x) tanh−1 1−y 13−15x 1 √3 − (1 − 5x) tanh−1 x − + 7 3 48y (1−y) (1−x−y) 1−x x (1−y) 2  4(1−y)−5x  4−5x 1 √ − 7√ 12y 1−x  8(1−y)2 −25x(1−y)+15x2 1 24y (1−y)3 (1−x−y)2

(1−y) 2

1−x−y

8−25x+15x2 (1−x)2




21 σ α1 α2 β1 β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)
√ x  1 √
(1−y)2 +6x(1−y)−15x2 tanh−1 1−y 1 √ 4 - 12 1 52 2 64xy − (1 + 6x − 15x2 ) tanh−1 x − 7 x (1−y) 2

− 72

1

3

2

1 3y [(1

4 - 12 1

7 2

2

5 √ 384x2 y x

4

- 12

4

1

4

4

1 2

1

- 21 2

4

1 2

1

1

4

1 2

1

4

1 2

1

2

2

4

1 2

1

5 2

2

4

1 2

1

3

2

4

1 2

1

7 2

4

1 2

1

4

4 1

1

- 21 2

4 1

1

1 2

2

4 1

1

3 2

2

1 24y 1 6y

2

2

2 105x3 y

1

(1−y)2 +2x(1−y)+5x2 (1−x−y) tanh−1

3 2

2




+ 3 + 4x − 15x2 3 2

2



5(1−y)−3x



(1−y)(1−x−y)

2 2

- 15x23 y

2

2

4 1

1

5 2

2

4 1

1

3

2

4 1

1

7 2

2

4 1

1

4

2

1

- 21 2

4

3 2 3 2

1

1 2

2

4

3 2

1

1

2

4

3 2

1

2

2

4

3 2

1

5 2

2

4

3 2

1

3

2

4

3 2

1

7 2

2

4

3 2

1

4

2

4 2

1

- 21 2





−

5−3x (1−x)2 4−3x

3

(1−x) 2

√

+

√3 x

√

x 1−y 5 (1−y) 2

tanh−1






3(1−y)2 +2x(1−y)+3x2 tanh−1 5 (1−y) 2

8(1−y)2 +4x(1−y)+3x2 5 (1−y) 2


√

1−x−y

 8(1−y)3 −80x(1−y)2 −39x2 (1−y)+6x3 1 24y (1−y)2 (1−x−y)4  8(1−y)2 +9x(1−y)−2x2 1 24y (1−y)2 (1−x−y)3 5(1−y)−2x





(1−y)(1−x−y) 2(1−y)−x (1−y)(1−x−y)


2 −


2 −

−

8+9x−2x2 (1−x)3

5−2x (1−x)2

2−x (1−x)2



+

√3 x

√

x 1−y

− tanh−1



√
x −

√
 x

1−y−3x (1−y)2 (1−x−y)

− (3 + 2x + 3x2 ) tanh−1

 √ − (8 + 4x + 3x2 ) 1 − x −

8−80x−39x2 +6x3 (1−x)4

(1−x−y) 2

(1−x) 2

(1−y) 2

3

3

− y)− 2 (1 − x − y)− 2 − (1 − x)− 2 ] √ x  3(1−y)−x √
 tanh−1 1−y 5 3−x 1 √ − tanh−1 x 3 16x2 y (1−y)(1−x−y) − 1−x − x (1−y) 2  8(1−y)2 −4x(1−y)−x2 8−4x−x2  2 − √1−x 3√ 3x3 y (1−y) 2 1−x−y  16(1−y)3 −272x(1−y)2 −659x2 (1−y)−30x3 2 −30x3 1 − 16−272x−659x 5 48y (1−y)(1−x−y) (1−x)5
√ x √
 5(1−y)+4x arcsin 3 x 1−y −105x 2 − (5+4x) arcsin 11 11 (1−x−y)

2

(1−x)

2

+

1−3x 1−x



√
x −3 √

x 1−y 9 (1−x−y) 2

arcsin

√ x √
 √ arcsin 1−y + 15 x − arcsin 7x 7 2 2 √ x (1−x−y) √
(1−x) arcsin arcsin x 1−y − 5 5


 x 2(1−y)+x
 1−x−y − x(x + 2) + 2 ln (1−x)(1−y) - 2x13 y (1−y)2  2 2 2 1 (1−y) −14x(1−y)−35x − 1−14x−35x 3y (1−x−y)5 (1−x)5   1−y+5x 1+5x 1 3y (1−x−y)4 − (1−x)4  8(1−y)2 +8x(1−y)−x2 8+8x−x2  1 − 7 3 7 24y (1−y) 2 (1−x−y) 2 (1−x) 2   4(1−y)−x 1 4−x 12y (1−y) 23 (1−x−y) 52 − (1−x) 52 √ x −1  √
 1−y 1−y+x 1+x 1 tanh 1 √ − tanh−1 x − − 3 8xy (1−y)(1−x−y)2 (1−x)2 x 3

3

− 105x 2

√ x  1 arcsin √ 1−y  arcsin x
1−y−2x 1 1−2x √ − − (1−y) 3 3 2 (1−x−y) + 1−x 8xy x (1−x−y) 2 (1−x) 2 1 −2 (1 − x − y)−1 − (1 − x)−1 ] 3y [(1 − y)
√
x  3 arcsin √ 1−y y 2x(2−y)+3(1−y)  arcsin x 5 √ √ √ − − 24x2 y (1−y)2 x 1−x−y 1−x

1 3y [(1

√
x

3




7 (1−y) 2

x  1 (1−y+3x) tanh−1 1−y 1 √ − (1 + 3x) tanh−1 5 16xy x (1−y) 2 1 1 1 − 52 (1 − x − y)− 2 − (1 − x)− 2 ] 3y [(1 − y)



− (1 + 2x + 5x2 )(1 − x) tanh−1

 + 1 − 15x

(1−x)


2 −

 4(1−y)−3x 1 12y (1−y) 25 (1−x−y) 32

5 √1 64x2 y x

1

x 1−y

8(1−y)2 +12x(1−y)+15x2 (1−x−y) 2

(8 + 12x + 15x )(1 − x) −   1−y−7x 1−7x 1 3y (1−x−y)4 − (1−x)4  8(1−y)2 −8x(1−y)+3x2 8−8x+3x2  1
52 − 5 24y 2 1 24y

√

7 (1−y) 2

(1−y)(1−x−y)

4 1

4

1

(1 − x − y) 2 − (1 − x) 2 ]


3(1−y)2 +4x(1−y)−15x2 (1−y)3

− - 12

− y)  3

(1−y−15x) (1−y)3

−

1+x+y (1−y)2

arcsin

√


 − (1 + x)

x


9 (1−x) 2

22 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)  16(1−y)2 +83x(1−y)+6x2 1

σ

α1 α2

β 1 β2

4

2

1

1 2

2

4

2

1

1

2

4

2

1

3 2

2

4

2

1

5 2

2

1−y+2x 1 16y (1−y)(1−x−y)2

4

2

1

3

2

1 3y [(1

− y)−1 (1 − x − y)−2 − (1 − x)−2 ]

2

5 16x2

2 1−y

1−x−y 1 1 2 1 x2 1−y + (1−x)(1−x−y) + x3 y ln (1−y)(1−x)   3 2 2 3 2 1 (1−y) −21x(1−y) −105x (1−y)−35x −35x3 − 1−21x−105x 3y (1−x−y)6 (1−x)6  2 2 2 1 (1−y) +10x(1−y)+5x − 1+10x+5x 3y (1−x−y)5 (1−x)5  8(1−y)2 +24x(1−y)+3x2 2 1 − 8+24x+3x 9 9 √ 24y 1−y(1−x−y) 2 (1−x) 2

4

2

1

7 2

4

2

1

4

2

4

5 2

1

- 12

2

4

5 2

1

1 2

2

4

5 2

1

1

2

4

1

3 2

2

4

5 2 5 2

1

2

2

4

5 2

1

3

2

4

5 2

1

7 2

2

4

5 2

1

4

2

4

3

1

- 21

2

(1−y)(1−x−y)4

48y

1−y+2x 1 3y (1−x−y)4



3

1

1 2

2

4

3

1

1

2

4

3

1

3 2

2

4

3

1

2

2

4

3

1

5 2

2

4

3

1

3

2

4

3

1

7 2

2

4

3

1

4

2

4

7 2

1

- 12

2

4

7 2

1

1 2

2

4

7 2

1

1

2

4

7 2

1

3 2

2

4

7 2

1

2

2

4

7 2 7 2

1

5 2

2

1

3

2

4

1+2x (1−x)4

 13(1−y)+2x 1 48y (1−y)(1−x−y)3 





+



−

13+2x (1−x)3

+

(1−y+4x) arcsin √3 7 x (1−x−y) 2

−

1+2x (1−x)2

−

(1−y−4x) arcsin √1 5 x (1−x−y) 2

1 (1−x)(1−x−y)

−

√1 xy




√ + 15 x

16+83x+6x2 (1−x)4

3(1−y)+4x arcsin

√




√

x 1−y

9 (1−x−y) 2

√

3(1−y)−4x arcsin

x 1−y

x 1−y

√

−

(1+4x) arcsin

−

(1−4x) arcsin

x 1−y

3 (1−x−y) 2



7 (1−x) 2

−

5 (1−x) 2

−

(3+4x) arcsin 9 (1−x) 2

√

x


√
 x

√
 x

(3−4x) arcsin

√

3 (1−x) 2




1−y+x 1 1+x 3y (1−x−y)4 − (1−x)4  4(1−y)+x  1 4+x 12y √1−y(1−x−y) 72 − (1−x) 72 5 5 1 − 21 (1 − x − y)− 2 − (1 − x)− 2 ] 3y [(1 − y) x  3 tanh−1 √ 1−y √
5 √ √ − tanh−1 x − 3(1−y)−5x 2 24x y (1−x−y)2 x 1−y

x






 2 2 −12x(1−y)+3x2 − 8−12x+3x - 3x23 y 8(1−y) 3 3 √ 1−y(1−x−y) 2 (1−x) 2  64(1−y)3 −1536x(1−y)2 −6885x2 (1−y)−2038x3 1 − 192y (1−x−y)6
√ 35(1−y)2 +56x(1−y)+8x2 arcsin

3

4

−

−

x

+

3−5x (1−x)2



64−1536x−6885x2 −2038x3 (1−x)6 √ (35+56x+8x2 ) arcsin x


1−y − −105x 2 13 13 (1−x−y) 2 (1−x) 2  2 2 2 64(1−y) +607x(1−y)+274x 1 − 64+607x+274x 5 192y (1−x−y)5
(1−x) √ x √
 2 √ 15(1−y)2 +40x(1−y)+8x2 arcsin 1−y +15 x − (15+40x+8x )11arcsin x 11 2 2 (1−x−y) (1−x)   2 2 1+4x+x2 1 (1−y) +4x(1−y)+x − 3y (1−x−y)5 (1−x)5
√ x √
 
2 3(1−y)2 +24x(1−y)+8x2 arcsin 11(1−y)+10x x 1−y 3 11+10x 1 √ 5 + − − (3+24x+8x ) arcsin 9 9 192y (1−x−y)4 (1−x)4 x (1−x−y) 2 (1−x) 2   1 2(1−y)+x 2+x 6y (1−x−y)4 − (1−x)4
√ x √
  1−y+14x 2 (1−y)2 −8x(1−y)−8x2 arcsin x 1−y 1 1+14x 1 √ − (1−8x−8x ) arcsin − − 7 7 3 3 64xy (1−x−y) (1−x) x

(1−x−y) 2

1 3y [(1

− x − y)−3 − (1 − x)−3 ]
√ x  1 3(1−y)2 −8x(1−y)+8x2 arcsin 1−y

5 √ 64x2 y x

5 (1−x−y) 2

−

(1−x) 2

(3−8x+8x2 ) arcsin 5 (1−x) 2

√

x


−3

1−y−2x (1−x−y)2


 x 2(1−y)−3x
 1−x−y - 2x13 y − x(2−3x) (1−x−y)2 (1−x)2 + 2 ln (1−x)(1−y)   4 3 2 2 3 4 2 1 (1−y) −28x(1−y) −210x (1−y) −140x (1−y)−7x −140x3 −7x4 − 1−28x−210x 3y (1−x−y)7 (1−x)7   3 2 2 3 2 1 (1−y) +15x(1−y) +15x (1−y)+x +x3 − 1+15x+15x 3y (1−x−y)6 (1−x)6
 √1−y 8(1−y)2 +40x(1−y)+15x2 2 1 − 8+40x+15x 11 11 24y (1−x−y) 2 (1−x) 2   5(1−y)2 +10x(1−y)+x2 5+10x+x2 1 − (1−x)5 15y (1−x−y)5
√  1−y 4(1−y)+3x  1 − 4+3x9 9 12y (1−x−y) 2 (1−x) 2   5(1−y)+x 5+x 1 15y (1−x−y)4 − (1−x)4 1 3y [(1

1

7

7

− y) 2 (1 − x − y)− 2 − (1 − x)− 2 ]

−

1−2x (1−x)2




23 σ

α1 α2

β1 β2

4

7 2

1

4

2

4

4

1

- 21

2

4

4

1

1 2

2

4

4

1

1

2

4

4

1

3 2

2

4

4

1

2

2

4

4

1

5 2

2

F2 (σ; α1 , α2 ; β1 , β2 ; x, y)
 √1−y 8(1−y)2 −20x(1−y)+15x2 2 5 (1−x−y) 2

15x3 y

8−20x+15x2

−

5 (1−x) 2



 128(1−y)4 −3968x(1−y)3 −26223x2 (1−y)2 −14702x3 (1−y)−280x4 2 1 −14702x3 −280x4 − 128−3968x−26223x 384y (1−x−y)7 (1−x)7
√ x 2 2 √ arcsin 3 (1−y) 35(1−y) +84x(1−y)+24x (35+84x+24x2 ) arcsin x
 1−y − −315x 2 15 15 (1−x−y) 2 (1−x) 2  3 2 2 3 2 128(1−y) +1779x(1−y) +1518x (1−y)+40x 1 +40x3 − 128+1779x+1518x 384y (1−x−y)6 (1−x)6
√ √ x 2 √ (1−y) 5(1−y)2 +20x(1−y)+8x2 arcsin 1−y arcsin x
 − (5+20x+8x )13 +105 x 13 (1−x−y) 2 (1−x) 2
2 2  2 (1−y) (1−y) +6x(1−y)+3x 1 − 1+6x+3x 3y (1−x−y)6 (1−x)6  2 2 2 113(1−y) +194x(1−y)+8x 1 − 113+194x+8x 384y (1−x−y)5
(1−x)5√ √ x 2 (1−y) (1−y)2 +12x(1−y)+8x2 arcsin arcsin x
 1−y − (1+12x+8x )11 + √15x 11 (1−x−y) 2 (1−x) 2   1 (1−y)(1−y+x) 1+x − (1−x)5 3y (1−x−y)5  2 2 2 3(1−y) +94x(1−y)+8x 1 − 3+94x+8x 384xy (1−x−y)4 (1−x)
4 √ √
 x 2 (1−y) (1−y)2 −12x(1−y)−24x2 arcsin 1−y − (1−12x−24x )9arcsin x − √3x 9 (1−x−y) 2

4

4

1

3

2

4

4

1

7 2

2

9 2

- 12

1

1 2

2

9 2

- 12

1

3 2

2

9 2

- 12

1

5 2

2

9 2 9 2

- 12 1 2

1

2

1

7 2 - 12

9 2

1 2

1

3 2

2

9 2

1 2

1

5 2

2

9 2

1 2

1

7 2

2

9 2

1

1

- 21

2

9 2

1

1

1 2

2

9 2

1

1

3 2

2

9 2 9 2

1

1

2

2

1

1

5 2

2

9 2

1

1

3

2

1

7 2

9 2

1

2

2

1 3y [(1

(1−x) 2

−4

−4

− y)(1 − x − y) − (1 − x) ]
x  3 (1−y) (1−y)2 −4x(1−y)+8x2 arcsin √ 1−y

5 √ 384x2 y x

7 (1−x−y) 2

2 + 3−10x−8x (1−x)3  5(1−y)3 −30x(1−y)2 +40x2 (1−y)−16x3 2 5 (1−y)4 (1−x−y) 2

35y

 15(1−y)2 −40x(1−y)+24x2 2 3 105y (1−y)4 (1−x−y) 2   5(1−y)−6x 2 5−6x √ √ 35y (1−y)4 1−x−y − 1−x 1 2 −4 (1 − x − y) 2 − 7y [(1 − y)  1−y−8x  2 1−8x 7y (1−x−y) 29 − (1−x) 92

−

−

−

(1−4x+8x2 ) arcsin 7 (1−x) 2

5−30x+40x2 −16x3 5 (1−x) 2

15−40x+24x2 3 (1−x) 2



√

x


−



1

(1 − x) 2 ]

 15(1−y)2 −20x(1−y)+8x2 2 2 − 15−20x+8x 5 5 105y 3 2 2 (1−y) (1−x−y) (1−x)  5(1−y)−4x  2 5−4x − 3 35y (1−y)3 (1−x−y) 32 (1−x) 2

1 1 2 −3 (1 − x − y)− 2 − (1 − x)− 2 ] 7y [(1 − y)  5(1−y)4 −60x(1−y)3 −90x2 (1−y)2 +20x3 (1−y)−3x4 2 5 35y (1−y) 2 (1−x−y)5

−

5−60x−90x2 +20x3 −3x4 (1−x)5

  5(1−y)3 +15x(1−y)2 −5x2 (1−y)+x3 2 5+15x−5x2 +x3 − 5 4 35y (1−x) (1−y) 2 (1−x−y)4  15(1−y)2 −10x(1−y)+3x2 2 2 − 15−10x+3x 5 3 105y (1−x) 3 2 (1−y) (1−x−y) 4 − 52 35xy (1 − x − y)  5(1−y)−3x 2 35y (1−y) 52 (1−x−y)2



2 8 105x2 y (1−x−y) 32 2 − 25 (1 − 7y [(1 − y)



9 2

1

1

4

2

4 √ 8 35x3 y 1−x−y

9 2

1

1

9 2

2

2 15x3 y

9 2

3 2

1

- 12

2

9 2

3 2

1

1 2

2

9 2

3 2

1

5 2

2

 

15

5

5

− (1 − x)− 2 − (1 − y)− 2 + 1  5−3x − (1−x) 2  2(1−y)+3x 2 − + 2 + 3x 3 − 5 (1−x) 2

8 − √1−x − √ −1 x

tanh

− (1 − x)−1 ]

8(1−y)2 +4x(1−y)+3x2 5 (1−y) 2

−1 1−y −tanh

√

x

√ x

 2 2 2 (1−y) −16x(1−y)−48x 11 7y 2 (1−x−y)  1−y+6x  2 1+6x 7y (1−x−y) 29 − (1−x) 92  5(1−y)−2x 2 35y (1−y)2 (1−x−y) 52

−



(1−y) 2

−1

x − y)



−

−

15(1−y)2 +5x(1−y)+3x2

1−16x−48x2

5−2x

5 (1−x) 2

11 (1−x) 2



+ 8 + 4x + 3x2

5 (1−y) 2





+ 15 + 5x + 3x2



3(1−y)2 −10x(1−y)−8x2 (1−x−y)3

24 σ

α1

α2

β1

β2

F2 (σ; α1 , α2 ; β1 , β2 ; x, y)

9 2

3 2

1

7 2

2

9 2

2

1

- 21

2

9 2

2

1

1 2

2

3 3 2 −2 (1 − x − y)− 2 − (1 − x)− 2 ] 7y [(1 − y)  4 3 2 2 3 4 2 (1−y) −20x(1−y) −90x (1−y) −20x (1−y)+x 3 7y (1−y) 2 (1−x−y)6

9 2

2

1

1

2

9 2

2

1

3 2

2

9 2

2

1

5 2

2

9 2

2

1

3

2

9 2 9 2

2

1

7 2

2

2

1

4

2

9 2

5 2

1

- 21

2

9 2

5 2

1

1 2

2

9 2

5 2

1

3 2

2

9 2

5 2

1

7 2

2

9 2

3

1

- 21

2

9 2

3

1

1 2

2

9 2

3

1

1

2

9 2

3

1

3 2

2

9 2

3

1

2

2

9 2

3

1

5 2

2

9 2

3

1

7 2

2

9 2

3

1

4

2

9 2

7 2

1

- 21

2

9 2

7 2

1

1 2

2

9 2

7 2

1

3 2

2

9 2

7 2

1

5 2

2

9 2

7 2

1

7 2

2

9 2

4

1

- 21

2

9 2

4

1

1 2

2

9 2

4

1

1

2 2 2

9 2

4

1

3 2

9 2

4

1

2

2

9 2 9 2

4

1

5 2

4

1

3

2

9 2

4

1

7 2

2

 5(1−y)3 +45x(1−y)2 +15x2 (1−y)−x3 2 3 35y (1−y) 2 (1−x−y)5   1 2(1−y)+5x 2+5x 7y (1−x−y) 92 − (1−x) 92  15(1−y)2 +10x(1−y)−x2 2 3 105y (1−y) 2 (1−x−y)4

−

+

2−5x 5 (1−x) 2

4(1−y)−5x 3







− x − y)−2 − (1 − x)−2 ] − (4 + x) −

1−20x−90x2 −20x3 +x4 (1−x)6

5+45x+15x2 −x3 (1−x)5

15+10x−x2 (1−x)4

  5(1−y)−x 2 5−x 35y (1−y) 23 (1−x−y)3 − (1−x)3  2 2(1−y)−5x 8 105x2 y (1−y) 32 − 2 − (1−x−y) 52 2 − 23 (1 7y [(1 − y)  4(1−y)+x 16 3 35x3 y (1−y) 2

−

−

 4−5x

+

3



(1−x−y) 2 (1−x) 2   3 2 2 3 2 (1−y) −24x(1−y) −144x (1−y)−64x 1−24x−144x2 −64x3 − 13 13 7y (1−x−y) 2 (1−x) 2   2 2 1+12x+8x2 2 (1−y) +12x(1−y)+8x − 11 11 7y (1−x−y) 2 (1−x) 2

 3(1−y)+4x  2 3+4x 21y (1−x−y) 29 − (1−x) 92 5 5 2 −1 (1 − x − y)− 2 − (1 − x)− 2 ] 7y [(1 − y)   4 3 2 2 3 4 2 2 (1−y) −28x(1−y) √−210x (1−y) −140x (1−y)−7x −140x3 −7x4 − 1−28x−210x 7y (1−x)7 1−y(1−x−y)7   3 2 2 +15x2 (1−y)+x3 2 (1−y) +15x(1−y) +x3 √ − 1+15x+15x 7y (1−x)6 1−y(1−x−y)6  8(1−y)2 +40x(1−y)+15x2 2 1 − 8+40x+15x 11 11 28y (1−x−y) 2 (1−x) 2   5(1−y)2 +10x(1−y)+x2 2 5+10x+x2 √ − (1−x)5 35y 1−y(1−x−y)5  4(1−y)+3x  4+3x 1 14y (1−x−y) 29 − (1−x) 92

  5(1−y)+x 2 5+x √ − (1−x) 4 35y 1−y(1−x−y)4 2 − 21 (1 − x − y)−3 − (1 − x)−3 ] 7y [(1 − y)  8(1−y)2 −20x(1−y)+15x2 8−20x+15x2 4 − 5 5 35x3 y (1−x−y) 2 (1−x) 2

1

− 8 (1 − y)− 2 − 1

 5(1−y)4 −160x(1−y)3 −1440x2 (1−y)2 −1280x3 (1−y)−128x4 2 15 35y (1−x−y) 2  5(1−y)3 +90x(1−y)2 +120x2 +16x3 2 13 35y (1−x−y) 2  15(1−y)2 +40x(1−y)+8x2 2 11 105y (1−x−y) 2   5(1−y)+2x 2 5+2x 35y (1−x−y) 29 − (1−x) 92

−

−

5+90x+120x2 +16x3 13 (1−x) 2

15+4−x+8x2 11 (1−x) 2



−




5−160x−1440x2 −1280x3 −128x4 15 (1−x) 2



7 2 − 27 − (1 − x)− 2 ] 7y [(1 − x − y)
  √1−y (1−y)4 −36x(1−y)3 −378x2 (1−y)2 −420x3 (1−y)−63x4 2 −420x3 −63x4 2 − 1−36x−378x 7y (1−x−y)8 (1−x)8
  √1−y (1−y)3 +21x(1−y)2 +35x2 (1−y)+5x3 2 2 +7x3 − 1+21x+35x 7y (1−x−y)7 (1−x)7  16(1−y)3 +120x(1−y)2 +90x2 (1−y)+5x3  2 1 +5x3 − 16+120x+90x 13 13 56y (1−x−y) 2 (1−x) 2

 √1−y




2 3(1−y)2 +10x(1−y)+3x2 − 3+10x+3x (1−x−y)6 (1−x)6  8(1−y)2 +12x(1−y)+x2 2 1 − 8+12x+x 11 11 28y (1−x−y) 2 (1−x) 2

2 21y


  √1−y 5(1−y)+3x 2 5+3x − (1−x) 5 35y (1−x−y)5  6(1−y)+x  6+x 1 21y (1−x−y) 92 − (1−x) 92 1 2 −4 2 − (1 − 7y [(1 − y) (1 − x − y)

x)−4 ]



25 σ α1 α2 β1 β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)  48(1−y)3 −231x(1−y)2 +280x2 (1−y)−105x3 2 1 −105x3 − 48−231x+280x 5 - 21 1 12 2 192y 4 (1−x−y)3 (1−y) (1−x)3 √ x √ √ tanh−1 1−y − tanh−1 x)] −105 x( 9 2 (1−y)  16(1−y)3 −72x(1−y)2 +90x2 (1−y)−35x3 2 3 1 − 16−72x+90x5 −35x 5 - 21 1 1 2 64y 5 9 (1−y) 2 (1−x−y) 2

1

3 2

2

5 - 21 1

2

2

- 21

5 2

5

5

- 21

1

5 - 21 1

3

5 - 21 1

7 2

5 - 21 1

2

2 2

1 384y

 24(1−y)2 −60x(1−y)+35x2 1 9 3 96y (1−y) 2 (1−x−y) 2

2

5

1 2

1

- 21 2

5

1 2

1

1

2

5

1 2

1

3 2

2

5

1 2

1

2

2

5

1 2

1

5 2

2

5

1 2

1

3

2

5

1 2

1

7 2

2

5

1 2

1

4

2

5 1

1

- 12 2

1

1 2

2

5 1

1

3 2

2

5 1

1

2

2

5 1

1

5 2

2

5 1

1

3

2

5 1

1

7 2

2

5 1

1

9 2

2

5 1

1

4

2

1

- 21 2

5

3 2 3 2

1

1 2

2

5

3 2

1

1

2

5

3 (1−x) 2

9 (1−y) 2 2

2

+100x(1−y)+105x − 3(1−y)(1−y) + 4 (1−x−y)   6(1−y)−7x 6−7x 1 −√ 9√ 24y 1−x (1−y) 2

5 √3 3072x2 y x






2

√

2

+

x 1−y

3+100x+105x2 1−x




9 (1−y) 2

+ 3 + 10x − 105x2

−

x 1−y

9 (1−y) 2

− (1 + 10x − 35x2 ) tanh−1

(1−y)3 +3x(1−y)2 +15x2 (1−y)−35x3 tanh−1

 33(1−y)2 −40x(1−y)+15x2 1
3 192y (1−y)(1−x−y)

√



1−x−y

+10x(1−y)−105x2 (1−y)4

(1−y−7x) tanh−1

15 √ x

√

x 1−y

33−40x+15x2 (1−x)3

+

− (1 − 7x) tanh−1



√

−1 x 1−y 15 tanh √ 7 x (1−y) 2

− tanh−1

 8(1−y)2 −12x(1−y)+5x2 1 7 5 32y (1−y) 2 (1−x−y) 2

√
 x

√
x

− (1 + 3x + 15x2 − 35x3 ) tanh−1

1 1 1 − 92 (1 − x − y) 2 − (1 − x) 2 ] 4y [(1 − y)   1−y−9x 1 1−9x 4y (1−x−y)5 − (1−x)5  16(1−y)3 −24x(1−y)2 +18x2 (1−y)−5x3 2 3 1 − 16−24x+18x7 −5x
72 64y 2 (1−x) (1−y)(1−x−y)

√
x

√
 x

2 − 8−12x+5x 5 (1−x) 2 x  3 (1−y+5x) tanh−1 √ 1−y 2 √
2 −22x(1−y)+15x2 1 √ + 3−22x+15x − (1 + 5x) tanh−1 x − 3(1−y) 7 3 (1−x−y)2 2 128xy (1−y) (1−x) x 2 (1−y)   6(1−y)−5x 6−5x 1 7 24y (1−y) 2 (1−x−y) 23 − (1−x) 32
√ x  3 2 √
(1−y)2 +2x(1−y)+5x2 tanh−1 +4x(1−y)−15x2 1−y 5 √ − (1 + 2x + 5x2 ) tanh−1 x − 3(1−y) 7 512x2 y (1−y)3 (1−x−y) x (1−y) 2 2 + 3+4x−15x 1−x 1 1 1 − 72 (1 − x − y)− 2 − (1 − x)− 2 ] 4y [(1 − y)  16(1−y)4 −208x(1−y)3 −165x2 +50x3 (1−y)−8x4 1 64y (1−y)3 (1−x−y)5 √ √ x 3 arcsin arcsin x
 1−y

−315x 2

5 1

24−60x+35x2

(1−y)2 +10x(1−y)−35x2 tanh−1

1 √3 512xy x



−

81−190x+105x (1−x)2

−

(1−y)4 (1−x−y)2

− 3(1−y)

4

(1−x) 2

 81(1−y)2 −190x(1−y)+105x2

(1−x−y)

11 2

−

(1−x)

1 12xy

−

16−208x−165x2 +50x3 −8x4 (1−x)5

11 2

 48(1−y)3 +87x(1−y)2 −38x2 (1−y)+8x3 1 192y (1−y)3 (1−x−y)4  33(1−y)2 −26x(1−y)+8x2 1
3 192y (1−y)(1−x−y)

−

−

33−26x+8x2 (1−x)3

48+87x−38x2 +8x3 (1−x)4

+

15 √ x

√

√ + 105 x

x 1−y 7 (1−x−y) 2

arcsin



−

arcsin

√

x 1−y 9 (1−x−y) 2

arcsin √
 x 7

(1−x) 2

 (1 − x − y)−3 − (1 − x)−3 − (1 − y)−3 + 1 √ x  3 arcsin √ 1−y 2 2 −14x(1−y)+8x2 arcsin x
1 √ + 3−14x+8x − − 3(1−y) 5 5 64xy (1−y)3 (1−x−y)2 (1−x)2 x (1−x−y) 2 (1−x) 2  3(1−y)−2x  1 3−2x 12y (1−y)3 (1−x−y)2 − (1−x)2 √
x  3 arcsin √ 1−y 2 2 +2x(1−y)−8x2 5 √ − arcsin 3x − 3(1−y) + 3+2x−8x 3 3 (1−x−y) 192x2 y (1−y) 1−x x (1−x−y) 2 (1−x) 2 √
x   15 arcsin √ 1−y 2 2 arcsin 7 √ √ − √1−x x − 15(1−y) +10x(1−y)+8x + 15 + 10x + 8x2 192x3 y (1−y)3 x 1−x−y 1 −3 (1 − x − y)−1 − (1 − x)−1 ] 4y [(1 − y)  2 2 2 1 (1−y) −18x(1−y)−63x − 1−18x−63x 4y (1−x−y)6 (1−x)6   1−y+7x 1+7x 1 4y (1−x−y)5 − (1−x)5  16(1−y)3 +24x(1−y)2 −6x2 (1−y)+x3 2 3 1 − 16+24x−6x9 +x 9 5 64y (1−y) 2 (1−x−y) 2 (1−x) 2

−

arcsin

√

x


9

(1−x) 2

26 σ

α1

α2

β1

β2

5

3 2

1

2

2

5

3 2

1

5 2

2

5

3 2

1

3

2

5

3 2

1

7 2

2

5

3 2

1

4

2

5

2

1

- 21

2

5

2

1

1 2

2

5

2

1

1

2

5

2

1

3 2

2

5

2

1

5 2

2

5

2

1

3

2

5

2

1

7 2

2

5

2

1

4

2

5

5 2

1

- 21

2

5

5 2

1

1 2

2

5

5 2

1

1

2

5

5 2

1

3 2

2

5

5 2

1

2

2

5

5 2

1

3

2

5

5 2

1

7 2

2

5

5 2

1

4

2

1

- 21

2

5

3

5

3

1

1 2

2

5

3

1

1

2

5

3

1

3 2

2

5

3

1

2

2

5

3

1

5 2

2

F2 (σ; α1 , α2 ; β1 , β2 ; x, y)  8(1−y)2 −4x(1−y)+x2 1 − 7 5 32y

8−4x+x2 7 (1−x) 2

(1−y) 2 (1−x−y) 2

 3(1−y)2 +8x(1−y)−3x2 1 64xy (1−y)2 (1−x−y)3 1 8y



2(1−y)−x

(1−y)(1−x−y)


52 −

1 4y [(1

3+8x−3x2 (1−x)3

−

2−x 5

(1−x) 2

 3(1−y)2 −2x(1−y)+3x2 5
2 128x2 y (1−y)(1−x−y) 5



−

√3 x

−

√

x 1−y 5 (1−y) 2

tanh−1

− tanh−1

 3−2x+3x2 (1−x)2

3

−

(1+x−y) tanh−1 √3 5 x (1−y) 2

√

x 1−y

√
 x

− (1 + x) tanh−1

3

− y)− 2 (1 − x − y)− 2 − (1 − x)− 2 ]

 32(1−y)4 −704x(1−y)3 −2553x2 (1−y)2 −260x3 (1−y)+20x4 1 6 128y
(1−y)2 (1−x−y) √ x √ 5(1−y)+6x arcsin 3 (5+6x) arcsin x
 1−y

−

√
 x

32−704x−2553x2 −260x3 +20x4 (1−x)6

− −315x 2 13 13 (1−x−y) 2 (1−x) 2  3 2 2 3 2 32(1−y) +247x(1−y) +40x (1−y)−4x 1 −4x3 − 32+247x+40x 128y (1−y)2 (1−x−y) (1−x)5 √ 5x √
 √ (1−y+2x) arcsin 1−y x − (1+2x) arcsin +105 x 11 11 (1−x−y) 2 (1−x) 2   1−y+3x 1+3x 1 4y (1−x−y)5 − (1−x)5 √ x √  81(1−y)2 +28x(1−y)−4x2 (1+6x) arcsin x
 1−y 1 81+28x−4x2 15 (1−y+6x) arcsin √ − − + 9 9 2 4 4 384y (1−y) (1−x−y) (1−x) x (1−x−y) 2 (1−x) 2 √ x √
  2 2 (1−y−6x) arcsin 2 3(1−y) +16x(1−y)−4x x 1−y 1 − (1−6x) arcsin − 3+16x−4x − √3x 7 7 128xy (1−y)2 (1−x−y)3 (1−x)3 2 2 (1−x−y) (1−x)   3(1−y)−x 1 3−x 12y (1−y)2 (1−x−y)3 − (1−x)3 √ x √
  3(1−y)2 −4x(1−y)+4x2 x 1−y 5 3−4x+4x2 3 (1−y−2x) arcsin √
− − (1−2x) arcsin − 2 5 5 128x2 y (1−x)2 x (1−x−y) 2

(1−y)(1−x−y)

(1−x) 2

1 −2 (1 − x − y)−2 − (1 − x)−2 ] 4y [(1 − y)   3 2 2 3 2 1 (1−y) −27x(1−y) −189x (1−y)−105x −105x3 − 1−27x−189x 4y (1−x−y)7 (1−x)7  3(1−y)2 +42x(1−y)+35x2 2 1 − 3+42x+35x 12y (1−x−y)6 (1−x)6  16(1−y)3 +72x(1−y)2 +18x2 (1−y)−x3  2 1 −x3 − 16+72x+18x 3 11 11 64y (1−y) 2 (1−x−y) 2 (1−x) 2

 3(1−y)+5x  1 3+5x 12y (1−x−y)5 − (1−x)5  24(1−y)2 +12x(1−y)−x2 1 9 3 96y (1−y) 2 (1−x−y) 2  6(1−y)−x 1 24y (1−y) 23 (1−x−y) 72

√

−

−

24+12x−x2

6−x

7 (1−x) 2

9 (1−x) 2





x  3 tanh−1 1−y 2 √
2 −8x(1−y)−3x2 5 √ − tanh−1 x − 3(1−y) + 3−8x−3x 3 3 3 192x2 y (1−y)(1−x−y) (1−x) x (1−y) 2 5 5 1 − 32 (1 − x − y)− 2 − (1 − x)− 2 ] 4y [(1 − y)  128(1−y)4 −3968x(1−y)3 −26223x2 (1−y)2 −14702x3 (1−y)−280x4 2 1 −14702x3 −280x4 − 128−3968x−26223x 512y (1−y)(1−x−y) (1−x)7
7 √ √ x 35(1−y)2 +84x(1−y)+24x2 arcsin 3 (35+84x+24x2 ) arcsin x
 1−y

−315x 2 − 15 15 (1−x−y) 2 (1−x) 2  3 2 2 3 2 128(1−y) +1779x(1−y) +1518x (1−y)+40x 1 +40x3 − 128+1779x+1518x 512y (1−y)(1−x−y)6 (1−x)6
√ x √ 2 √ 5(1−y)2 +20x(1−y)+8x2 arcsin arcsin x
 1−y +105 x − (5+20x+8x )13 13 (1−x−y) 2 (1−x) 2   2 2 1+6x+3x2 1 (1−y) +6x(1−y)+3x − (1−x)6 4y (1−x−y)6  2 2 113(1−y) +194x(1−y)+8x2 1 − 113+194x+8x 512y (1−y)(1−x−y)5 (1−x)5
√ x √ 2 (1−y)2 +12x(1−y)+8x2 arcsin arcsin x
 1−y − (1+12x+8x ) 11 + √15x 11 (1−x−y) 2 (1−x) 2   1−y+x 1+x 1 − 4y (1−x−y)5 (1−x)5  3(1−y)2 +94x(1−y)+8x2 2 1 − 3+94x+8x 4 512xy (1−y)(1−x−y)4
(1−x)√ √
 x 2 (1−y)2 −12x(1−y)−24x2 arcsin 1−y − (1−12x−24x )9arcsin x − √3x 9 (1−x−y) 2

(1−x) 2

27 σ α1 α2 β1 β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)
√ x  3 (1−y)2 −4x(1−y)+8x2 arcsin 1−y 7 5 √ 5 3 1 2 − 7 2 512x2 y x (1−x−y) 2  2 2 2 −10x(1−y)−8x + 3+94x+8x − 3(1−y) (1−y)(1−x−y)3 (1−x)3 5 3

1

4

5

7 2

1

- 12 2

5

7 2

1

1 2

2

5

7 2

1

1

2

5

7 2

1

3 2

2

5

7 2

1

2

2

5

1

5 2

2

5

7 2 7 2

1

3

2

5

7 2

1

4

2

5 4

1

- 21 2

5 4

1

1 2

2

2

5 4

1

1

2

5 4

1

3 2

2

(1−4x+8x2 ) arcsin

1

2

2

5 4

1

5 2

2

5 4 5 4

1

3

1

7 2

2 2

5(1−y)+3x 1 5+3x 20y (1−x−y)5 − (1−x)5  6(1−y)+x 1 6+x 24y √1−y(1−x−y) 92 − (1−x) 29





1 4y [(1

1

7



7

− y)− 2 (1 − x − y)− 2 − (1 − x)− 2 ]  256(1−y)4 −10240x(1−y)3 −99021x2 (1−y)2 −102592x3 (1−y)−13628x4

1 1024y (1−x−y)8 256−10240x−99021x2 −102592x3 −13628x4 − (1−x)8
√ x 3 105(1−y) +378x(1−y)2 +216x2 (1−y)+16x3 arcsin 3 1−y

−315x 2 17 (1−x−y) 2  256(1−y)3 +5175x(1−y)2 +8132x2 (1−y)+1452x3

1

4

2

√ +35 x 15 (1−x−y) 2   3 2 2 3 2 1 (1−y) +9x(1−y) +9x (1−y)+x +x3 − 1+9x+9x 4y (1−x−y)7 (1−x)7
 7 33(1−y)2 +104x(1−y)+28x2 2 ) 1 − 7(33+104x+28x 1024y (1−x−y)6 (1−x)
6 √ + √5x  3(1−y)2 +6x(1−y)+x2 1

(105+378x+216x2 +16x3 ) arcsin 17 (1−x) 2

−

x 1−y

13 (1−x−y) 2

2 − 3+6x+x (1−x−y)6 (1−x)6  (1−y)2 +68x(1−y)+36x2 2 3 − 1+68x+36x 1024xy (1−x−y)5 (1−x)5
√ x (1−y)3 −18x(1−y)2 −72x2 (1−y)−16x3 arcsin 1−y − √1x 11 (1−x−y) 2   3(1−y)+x 1 3+x 12y (1−x−y)5 − (1−x)5

5 √3 3072x2 y x



2

−

(1−x)

15 2

(5+90x+120x2 +16x3 ) arcsin 13 (1−x) 2



12y

1 4y [(1

−

√
 x

2 +1452x3 − 256+5175x+8132x (1−x−y)7 (1−x)7
√ x √ 35(1−y)3 +210x(1−y)2 +168x2 (1−y)+16x3 arcsin (35+210x+168x2 +16x3 ) arcsin x
 1−y

1 1024y




(1−y)3 −6x(1−y)2 +24x2 (1−y)+16x3 arcsin 2

−16x(1−y)−92x − 3(1−y) (1−x−y) + 4

5 4

√
x

1 −1 (1 − x − y)−3 − (1 − x)−3 ] 4y [(1 − y)   4 3 2 2 3 4 2 −420x3 −63x4 1 (1−y) −36(1−y) −378x (1−y) −420x (1−y)−63x − 1−36−378x 4y (1−x−y)8 (1−x)8   3 2 2 3 2 +7x3 1 (1−y) +21x(1−y) +35x (1−y)+7x − 1+21x+35x 4y (1−x−y)7 (1−x)7  16(1−y)3 +120x(1−y)2 +90x2 (1−y)+5x3  2 +5x3 1 − 16+120x+90x 13 13 √ 64y 1−y(1−x−y) 2 (1−x) 2  3(1−y)2 +10x(1−y)+3x 3+10x+3x2  1 − (1−x)6 12y (1−x−y)6  2 2 2 8(1−y) +12x(1−y)+x 1 − 8+12x+x 11 11 √ 32y 1−y(1−x−y) 2 (1−x) 2

5(1−y)3 +90x(1−y)2 +120x2 (1−y)+16x3 arcsin

5 4

7 (1−x) 2

9 (1−x−y) 2

3−16x−92x2 (1−x)4

− x − y)−4 − (1 − x)−4 ]



√

− x 1−y

(1−18x−72x2 −16x3 ) arcsin 11 (1−x) 2

−

√

x


(1−6x+24x2 +16x3 ) arcsin 9 (1−x) 2

√
 x

√

x


28 REPRESENTATIONS OF APPELL HYPERGEOMETRIC FUNCTION F2

III.

σ

α1

α2 β1

β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)

σ

α1

1

β1

2

a+1

a−

1 2

1

2a

2

a+1

a+

1 2

1

2a

2

a+1

a+

1 2

1

2a + 1 2

a+1

a+

1

a+

1

1 2 3 2

2

a+1

1 2 1 2

a+1

a 2

+1 1

a 2

2

a+1

a

1

a+1 2

a

a

1

a+1 2 2

2

a+1

−a

1

1 2

a+1

1−a 1

1 2

2

a+1

1−a 1

3 2

2

a+1

2−a 1

3 2

2

a+1

α

1

a

2

a+1

α

1

α

2

a+1

1

1

2

2

a+

3 2

a

1

2a

2

a+

3 2

a

1

2a + 1 2

2a + 1 a + 1 1

a

2

a 2 1 2

+2 a

2

n+1 2

1

a 2 n−1 2

1

0

1

β

2

1

α1

1

α1

2

1

β+1 1

β

2

2

b

1

2

2

1

1

1

m

2

3 2

1

1

m+

2

n

1

m

2

2

n

1

m

2

2

n

1

n+

1 2

2

2

n−

1

n+

1 2

2

2

n

1

n+1 2


y x , 1−x , (σ 6= 0) (1 − x)−σ F2 σ; β1 − α1 , 1; β1 , 2; x−1 √ √ √ 
 1−y+ 1−x−y
1−2a 1+ 1−x 1−2a 1 √1 − ay 2 2 1−y
2a−1 
2a−1  1 1 √ 1 2 √ √ √2 − √1−x ay 1−x−y 1−y+ 1−x−y 1+ 1−x 
2a 
2a 2 1 √ √ , (a 6= − 12 , −1) − 1+√21−x ay 1−y+ 1−x−y

√ √ √ √ √ √ ( 1−y+ x)−2a +( 1−y− x)−2a −(1+ x)−2a −(1− x)−2a 2ay  √ √ 1−2a √ √ √ √ 1 √ 1 − y − x) − (1 − x)1−2a − ( 1 − y + x)1−2a + (1 + x)1−2a ( 2a(2a−1) xy   1−y+x 1 1+x ay (1−x−y)a+1 − (1−x)a+1   a a+1 Γ(a)Γ(1 − a), x = y = 12 , a < 1, a 6= 0, −1, −2, . . . -2a ψ( a+1 2 ) − ψ( 2 ) + 2 a 1 -2a (Γ(a + 1)Γ(1 − a) + a[ψ( a2 ) − ψ( a−1 2 )] + 1−a ), x = y = 2 , a < 2, a 6= 0, −1, −2, . . .

√




√
 − cos (2a) arcsin x √ x
√
 cos (2a−1) arcsin x  1−y 1 cos (2a−1) arcsin √ , (a 6= 0) − 1√ ay 1−x (1−y)a− 2 1−x−y
x  sin (2a−1) arcsin √ 1−y √
 1 √ − sin (2a − 1) arcsin x , (a 6= 0) 1 a− a(2a−1) xy 2 (1−y)
√
x  sin 2(a−1) arcsin √ 1−y sin 2(a−1) arcsin x  1 √ √ √ − 2a(a−1) xy (1−y)a−1 1−x−y 1−x   1 α−a −α −α (1 − x − y) − (1 − x) , (a 6= 0) ay (1 − y)   1 −a −a − (1 − x) ay (1 − x − y)  1 cos (2a) arcsin ay (1−y)a

x 1−y

(1−x−y)1−a −(1−y)1−a −(1−x)1−a +1 a(a−1)xy


2a−1 
2a−1 2 1 1 √ √ √2 − √1−x 1−y+ 1−x−y 1+ 1−x (1−x−y)(1−y)
2a  1
2a  1 2 √ √ √ − 1+√21−x (a+ 12 )y 1−y 1−y+ 1−x−y   1−y+x 1 1+x 2ay (1−x−y)2a+1 − (1−x)2a+1 1 (a+ 12 )y



√

a

1

1 2

2

1 2

2

  (1−y) 2 −1 (1−y+x) 1+x 2 − (1−x) a+1 (a+4)y (1−x−y)a+1  (n−1)−nx (n−1)(1−y)−nx  2 √ √ , n = 2, 3, . . . − (n−1)y 1−x−y 1−x ln(1−y) - y
1 1−x y ln 1−x−y 1 1 1−x x β (x−1)(x+y−1) + y ln( 1−x−y )  1−x−y
1−b  1 − (1 − x)1−b (b−1)xy 1−y  Pm (x+y−1)k−2 −(x−1)k−2
m−1
1 − x+y−1 ln 1−x−y k=3 y x 1−y (m−k+1)xk−2


x−1 m−1 x

ln(1 − x)  − ln(1 − y) , (m = 1, 2, 3...) q  √
√ 2( 21 )m −1 −1 x m−1 m−1 tanh tanh x (m−1)!xm y 2 x (x + y − 1) 1−y − (x − 1)
k √ Pm−1 + k=1 (k−1)!x 1 − y(x + y − 1)m−k−1 − (x − 1)m−k−1 ( 12 )k  Pn−m (m−n)k xk  (1−x−y)m−n−1 m−1 − (1 − x)m−n−1 (m ≤ n) k=0 k!(m+k−1) y (1−y)m−n+k  Pm−n−1 (m−n−k−1)!
(m−1)! k−1 − (x − 1)k−1 k=1 (m−n−1)!xy (m−k−1)!xk−1 (x + y − 1) k−1 m−n−1 P n−1 (1−y) (x+y−1) −(x−1)m−n−1 − (n−1)!x1m−n−2 k=1 (n−k)xk
 m−n−1 +x−n ((x + y − 1)m−n−1 (1 − y)n−1 ln( 1−x−y ln(1 − x)) , 1−y ) − (x − 1) (m > n) q √

  √ (1−y)n−1 k P ( 12 )n n−2 (k−1)!x x arcsin x √ (1 − y)n−k−1 − 1 − √1−x − k=1 ( 1 ) arcsin 1−y (n−1)!xn y 2 x 1−x−y 2 k q
Pn−1 xk 
 −1 −1 √ 2n−1 √ x n− 23 n−k−1 x (1 − y) tanh x − k=1 −1 n x y 1−y − tanh 2k−1 (1 − y) 

 Pn−2 k - xnn y (1 − y)n−1 ln 1−x−y − ln(1 − x) + k=1 xk (1 − y)n−k−1 − 1 , (n = 1, 2, 3, ...) 1−y +

29 σ

α1

α2 β1

2

1

1

m

2

1 2

1

m+

1 2

2

1

1

m+

1 2

2

-n

1

m

2

-n

1

-m

n+1 1

1

m

n+1 1

1

m

n+1 1

1

m

n+1 1

1

n+

1 2

1

1

n+

1 2

n+1 1

1

n+1

1−n 1

1

-m

1−n 1

1

m

1

-n

n+

1−

1 2

n 2

3−n 2

1−n 2

- n2

1

-n

β2 F2 (σ; α1 , α2 ; β1 , β2 ; x, y)  Pm−1 (x+y−1)k−2 −(x−1)k−2 2 (m−1)x k=2 y (m−k)xk

 −x−m (x + y − 1)m−2 ln 1−x−y − (x − 1)m−2 ln(1 − x) 1−y x  √ (x+y−1)m−1 tanh−1 √ 1−y √
( 21 )m √ 2 (m−1)!x x − (x − 1)m−1 tanh−1 x my 2 1−y
k Pm−2 + k=1 (k−1)!x (x + y − 1)m−k−1 − (x − 1)m−k−1 ( 12 )k √ x √  √ arcsin 1−y arcsin x
m−1 (−1) 2 2m−1 x 3 −m − 3 −m xm y (1−x−y) 2 (1−x) 2
 Pm−1 xk (x + y − 1)m−k−1 − (x − 1)m−k−1 , (m = 1, 2, 3, ...) + k=1 2k−1
 Pm−3 (1−m−n)k xk n!(−x)1−m  (1−x−y)m+n−1 m+n−1 − (1 − x) − k=0 (1 − y)m−k−2 + 1 2 (m)n y (1−y)n+1 k! 
Pm−n (n−m)k (1−y)k 1 2 y1 m+1 k=0 (n+2)k (1−x−y)k+1 − (1−x)k+1 n+1
 n (1−y)−n−1 n!xm+1 1 (n > m) + (−1)(−m) (x+y−1)m−n+1 − (x−1)m−n+1 , n  k  m−n−1 P n−m (m−n)k x (1−x−y) m−n−1 , (m ≤ n, n 6= 0) 2 m−1 k=0 k!(m+k−1) ny (1−y)m+k−1 − (1 − x)
 Pm−n−1 (m−n−k−1)! (x+y−1)k−1 (m−1)! k−1 2 (m−n−1)!nxy k=1 (1−y)n−1 − (x − 1) (m−k−1)!xk−1 k−n m−n−1 m−n−1 P n−1 (1−y) (x+y−1) −(x−1) − (n−1)!x1m−n−2 k=1 (n−k)xk
 m−n−1 ln(1 − x)) , (m > n, n 6= 0) +x−n ((x + y − 1)m−n−1 ln( 1−x−y 1−y ) − (x − 1)

(1−m)n  (−x)1−m (1 − x)m−n−1 ln(1 − x) − (1 − x − y)m−n−1 ln 1−x−y 2 (n−1)!ny 1−y
 Pm−n−1 (n−m+1)k Pn+k−1 xj−k−1 (1 − y)k−j − 1 , (m > n > 1) +(−x)1−n k=0 j=1 k! j √
x
  √ arcsin √ 1−y k Pn−1 ( 12 )n x √ 2 (n−1)!nx x √1−x−y − arcsin (1 − y)−k − 1 , (n 6= 0) − k=1 (k−1)!x ny 2 ( 12 )k 1−x q 

 P √ √ k 1 n−1 x x 2 −k − 1 − tanh−1 x − k=1 2 xn2 y x tanh−1 1−y , (n = 1, 2, 3, ...) 2k−1 (1 − y) 

 P k n−1 1−x−y + k=1 xk (1 − y)−k − 1 , (n = 1, 2, 3, ...) 2 - xn1 y ln (1−y)(1−x) 
k+n+1 P m−n (n−m)k (1−y) 1 m+1 1 2 - ny k=0 (n+2)k (1−x−y)k+1 − (1−x)k+1 n+1
 m+1 n m+1 (1−y) n!x 1 (n > m) + (−1)(−m) (x+y−1)m−n+1 − (x−1)m−n+1 , n (n−1)!(−x)1−m  (1 − x − y)m+n−1 − (1 − x)m+n−1 2 (m)n y
 Pm−2 (1−m−n)k xk − k=0 (1 − y)n−k − 1 k! √ √ √ √ √ √  n+1 n+1 −( 1−y− 1−x−y)n+1 −(1− 1−x)n+1  , 2 - 2n1ny ( 1−y+ 1−x−y) √1−x−y − (1+ 1−x) √1−x 2

2

b

1

b−m 2

2

b

1

m

2

2

b

1

m

2

b+1

1

1

b−m 2

b+1

1

1

m

2

b+1

1

1

2

2

b+1

-n

1

c

2

 (√1−y+√1−x−y)n+1 −(√1−y−√1−x−y)n+1 1 √ 2n (1−n)y (1−y)(1−x−y)

(n 6= 0)

−

√ √ (1+ 1−x)n+1 −(1− 1−x)n+1  √ , 1−x

(n 6= 0, 1)


 
k−b+1 (1−y)m−b+1 1 b−m−1 Pm−1 (−m)k (1−y) 1 1 m! k=0 (2−b)k (1−x−y)k+1 − (1−x)k+1 − (1−b)m (x+y−1)m+1 − (x−1)m+1 by b−1  (m−1)! m−b−1 (1 − y)b+1 − (1 − x)m−b−1 (1−b)m−1 (−x)m−1 y (1 − x − y)
 P (b−m−1)k xk − m−2 (1 − y)m−k − 1 , (m = 1, 2, 3, ...; m − b 6= 1, 2, 3, ...) k=0 k!  1−x−y
1−b  1 1−b − (1 − x) (b−1)xy 1−y
 
(1−y)m (1−y)k 1 1 m! 1 b−m−1 Pm−1 (−m)k k=0 (2−b)k (1−x−y)k+1 − (1−x)k+1 − (1−b)m (x+y−1)m+1 − (x−1)m+1 , y b−1  (m−1)! m−b−1 − (1 − x)m−b−1 b(1−b)m−1 (−x)m−1 y (1 − x − y)
 k Pm−2 kx (1 − y)m−b−k−1 − 1 , (m = 1, 2, 3, ...; m − − k=0 (b−m−1) k!   1 1−b 1−b 1−b − (1 − x) − (1 − y) +1 b(b−1)xy (1 − x − y)
Pn (−n)k (b)k xk 1 (1 − y)−(b+k) − 1 k=0 by (c)k k! 
n x n+c−1 n+c−b x b−c d x1−c (1−x−y) [( 1−y ) (1 − 1−y ) ] = b(c) x (1−y)n+1 d( 1−y ) ny
 n d −(1 − x)n+c−b dx [xn+c−1 (1 − x)b−c ]

(b 6= 0)

b 6= 1, 2, 3, ...)

30 σ

α1

α2

β1

β2

1−n

b

1

c

2

F2 (σ; α1 , α2 ; β1 , β2 ; x, y)
k 1 Pn−1 (−n)k (b)k x (1 − y)n−k − 1 - ny k=0 (c)k k! 
n x n+c−1 n+c−b x b−c d x1−c (1−x−y) (1 − 1−y ) ] [( 1−y ) = − n(c) x d( 1−y ) (1−y)1−b ny
 n n+c−1 b−c n+c−b d [x (1 − x) ] , (n 6= 0) −(1 − x) dx

Acknowledgments

The present study was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant GP249507 (NS).

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