Reduction formulas for the Appell and Humbert functions

Integral Transforms and Special Functions

ISSN: 1065-2469 (Print) 1476-8291 (Online) Journal homepage: http://www.tandfonline.com/loi/gitr20

Reduction formulas for the Appell and Humbert functions Yu. A. Brychkov To cite this article: Yu. A. Brychkov (2016): Reduction formulas for the Appell and Humbert functions, Integral Transforms and Special Functions, DOI: 10.1080/10652469.2016.1249481 To link to this article: http://dx.doi.org/10.1080/10652469.2016.1249481

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Date: 11 November 2016, At: 22:50

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS, 2016 http://dx.doi.org/10.1080/10652469.2016.1249481

Reduction formulas for the Appell and Humbert functions Yu. A. Brychkov Dorodnicyn Computing Center, Federal Research Center ‘Computer Science and Control’ of the Russian Academy of Sciences, Moscow, Russia ABSTRACT

ARTICLE HISTORY

Representations are obtained for some special cases of the Appell functions F1 , F3 and the Humbert functions 1 , 2 , 1 , 2 in terms of elementary and simpler special functions.

Received 4 August 2016 Accepted 12 October 2016 KEYWORDS

Appell functions; Humbert functions; special functions; reduction 2010 MATHEMATICS SUBJECT CLASSIFICATION

C33; 33C77; 33C70

1. Introduction In recent years, the double hypergeometric functions (the Appell functions) and their confluent forms (the Humbert functions) are increasingly used in various applications, for example in theoretical physics [1,2] and in communication theory [3–6]. Reduction of these functions for specific values of parameters and variables is of interest, especially in connection with simplification algorithms in computer algebra systems [1]. Below, we give a number of reduction formulas for the Appell functions 

F1 (a, b, b ; c; w, z) =

∞ ∞   (a)k+l (b)k (b )l wk zl k=0 l=0

F3 (a, a , b, b ; c; w, z) =

(c)k+l

k!l!

∞ ∞   (a)k (a )l (b)k (b )l wk zl k=0 l=0

(c)k+l

|w|, |z| < 1,

,

k!l!

|w|, |z| < 1,

,

(1.1)

(1.2)

and for the confluent Appell functions (the Humbert functions) 2 (b, b ; c; w, z) =

∞  ∞  (b)k (b )l wk zl k=0 l=0

3 (b; c; w, z) =

[email protected]

© 2016 Informa UK Limited, trading as Taylor & Francis Group

k!l!

∞ ∞   (b)k wk zl , (c)k+l k!l! k=0 l=0

CONTACT Yu. A. Brychkov

(c)k+l

,

(1.3)

(1.4)

2

YU. A. BRYCHKOV

∞ ∞   (a)k (a )l (b)k wk zl

1 (a, a ; b; c; w, z) =

(c)k+l

k=0 l=0

2 (a, b; c; w, z) =

∞ ∞   (a)k (b)k wk zl k=0 l=0

(c)k+l

k!l!

|w| < 1

k!l!

,

,

|w| < 1.

(1.5)

(1.6)

In what follows, n denotes a non-negative integer in all formulas.

2. Reduction formulas for F1 For derivation of special cases of F1 , we will use the reduction formula     −a  w−z F1 (a, b, b ; b + b ; w, z) = (1 − z) 2 F1 a, b; b + b ; 1−z

(2.1)

[7, p.245,(28)] and the differentiation formulas 



Dnz [zb +n−1 F1 (a, b, b ; c; w, z)] = (b )n zb −1 F1 (a, b, b + n; c; w, z),

(2.2)

Dnz [za+n−1 F1 (a, b, b ; c; wz, z)] = (a)n za−1 F1 (a + n, b, b ; c; wz, z)

(2.3)

[8,(4.2–3)]. (a) Substituting a = b = b = 1 into (2.1) we obtain F1 (1, 1, 1; 2; w, z) =

ln(1 − z) − ln(1 − w) . w−z

(2.4)

Applying the formulas (2.2), (2.3) consequently, using the symmetry relation F1 (a, b, b ; c; w, z) = F1 (a, b , b; c; z, w) and the formula Dnz [zα (a − z)β ] (α,β)

where Pn

n α−n

= n!a z

(α−n,β−n) (a − z)β−n Pn

 2z , 1− a



(2.5)

(z) is the Jacobi polynomial, we yield the equality

F1 (m + 1, n + 1, p + 1; 2; w, z) =

m   n k  (−1)n+1 wn+p   z m − 1 r m(z − w)n+p+1 w r=1 k=0

s   r z n Ps(r−s,−r−s) (1 − 2z) k 1−z s=0  δs,0 wr (n−k+r,−k−r) − P (1 − 2w) , (2.6) (1 − w)k+r k  p

×

where δm,n is the Kronecker symbol.

w−z w(1 − z)

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

3

(b) Substituting a = b = 1, b = 12 into (2.1) we obtain √   1 w−z 1 3 F1 1, , 1; ; w, z = √ arctanh √ . √ 2 2 1−z w−z 1−z Application of the formula (2.3) gives √   (1/2)n (1 − z)−n−1/2 w−z 1 3 F1 n + 1, , 1; ; w, z = arctanh √ √ 2 2 n! w−z 1−z   n (1 − z)−n−1  1 w(1 − z) k (k−1/2,k−n−1/2) Pn−k (1 − 2z) + 2w k 1−w k=1  

 k−1   1−w r 2 2z − 1 r/2 , (2.7) (z − z) Pr √ × 1+ w(1 − z) z2 − z r=1 where Pn (z) is the Legendre polynomial. Applying the differentiation formulas (2.2) with respect to the second and third parameters, and the formula Dnz [zc−1 F1 (a, b, b ; c; wz, z)] = (−1)n (1 − c)n zc−n−1 F1 (a, b, b ; c − n; wz, z)

(2.8)

[8,(4.4)] for the fourth parameter, one can derive a representation for the function F1 (m + 1, n + 12 , p + 1; r + 32 ; w, z), where m, n, p, r are positive integers. Similar results can be obtained a = 1/2, b = 1/2, b = 1. Substituting a = m + 3/2, b = 1, c = 3/2, into the integral representation [8, (3.3)] F1 (a, b, b ; c; w, z) =

(c) (1 − w)c−a−b (b)(c − b)  1 × xc−b−1 (1 − x)b−1 (1 − wx)a−c 2 F1 (a, b ; c − b; zx) dx,

(2.9)

0

and making use of the relations (2.2) and (2.5), we obtain   3 3 F1 m + , n + 1, a; ; w, z = (1 − z)−m−n−1 2 2 ⎞ ⎛ 3 1 m   k , m + , a k +  m (−w) (k,−m−n−1) ⎜ 2 2 ⎟ P × (1 − 2w)3 F2 ⎝ ⎠. 3 1 k 2k + 1 n k + , ;z k=0 2 2 Substituting a = −(m + 1)/2, b = 1, c = 3/2 into the integral representation (2.9) and using (2.2) we get  [(m+1)/2]  m + 1 3 m m+1 (m+1)/2−n wk F1 − , − , n + 1; ; w, z = (1 − z) 2k 2 2 2 k=0   ⎞ ⎛ m+3 1 m+3 (z − z2 )p n , + p k + p +  2 p (p,(m+1)/2−n+p)) ⎟ ⎜ 2 2 Pn−p × (1 − 2z)2 F1 ⎝ ⎠. 3 p!(2k + 2p + 1) p=0 k + p + ;z 2 

4

YU. A. BRYCHKOV

(c) Other formulas which can be derived from (2.9) have the form   m   3 (m − 1)!  m 1 (1/2−k,−k) 1 {w(1 − w)−k Pk−1 (1 − 2w) F1 m + , 1, 1; ; w, z = 2 2 2(w − z) 2 m k=1 k (1/2−k,−k)

(1 − 2z)} [m ≥ 1], − z(1 − z)−k Pk−1    m   5 3(m − 1)! 3 m+1 1 {(mw − k)(1 − w)−k−1 F1 m + , 1, 1; ; w, z = 2 2 4(w − z) 2 m+1 k=0 k + 1 (−k−1/2,−k−1)

(−k−1/2,−k−1)

× Pk (1 − 2w) − (mz − k)(1 − z)−k−1 Pk (1 − 2z)} [m ≥ 1],     √ √ 3 1 2z , 1, n + 1; ; w, z = wn (w − z)−n−1 w arctanh wPn(0,−n−1) 1 − F1 2 2 w √     n √ −n−1/2 z1 w−z k z z +√ − arctanh zC2n+1 w 2 k w(1 − z) z/w k=1    2z (k+1/2,k−n−1) (1/2−k,−k) Pk−1 × Pn−k 1− (1 − 2z) , w   1 3 m!n!(m + q − 1)! (1 − w)1/2−n 1 F1 m + ; n + , p + 1; − q; w, z = (−1)q+1  1   1   1  2 2 2 w(1 − z) 2 m 2 n −2 q  1 m+q k w k1 1 −1  − 2 k −k  1 − w k2 ( 1−w ) 1 2 × (k1 − q)!(m + q − k1 )! (k1 − k2 − 1)! w(1 − z) k1 =q

k2 =0

(k −k2 −3/2,k2 −k1 −n+1/2)

× Pn 1

(1 − 2w)

k3 p    z p k3 1−z

k3 =0 (k −k −1/2,−k2 −k3 −1) × Pk23 2 (1 − 2z)

[q ≥ 1].

One more reduction formula for F1 was derived in [9] √  1 π (2 − a) 1 3 (−w)a F1 a, 1, a − ; ; w, z = 2 2 2(a − 1)w ( 32 − a)  √ −2a 1/2−a × [(w − 1)(w − z)] − u+ (1 + z) 2 F1 a, 1; 2 − a; 

 √ −2a −u− (1 − z) 2 F1 a, 1; 2 − a;

where u± = w ±

√ √ z + (w − 1)(w − z).

u2− √ w(1 − z)2



,

u2+ √ w(1 + z)2



INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

5

3. Reduction formulas for F3 (a) The expansion F3 (a, a , b, b ; c; w, z) =

∞  (a )k (b )k k=0

(c)k k!

zk 2 F1 (a, b; c + k; w)

(3.1)

follows from the definition (1.2) after evaluation of one of the sums. Substituting b = n−a and w = 1/2 and making use of the relation   n    1 n n−b √ (b) = 2 F π a, n − a; b; (−2)−m 2 1 2 (−a)n m=0 m   1 1 × (a + b − n)m + ( a+b+m−n )( b−a+m−n+1 ) ( a+b+m−n+1 )( b−a+m−n ) 2 2 2 2 [10, 8.1.1.131], we obtain     n  2a−1 (c) 1 n (−1)k F3 a, a , n − a, b ; c; , z = k 2 (a + c − n)(−a)n k=0     ((k + a + c − n)/2) (2, a ), (2, b ), (k + a + c − n)/2 × F 5 4 (2, a + c − n), (k − a + c − n)/2, 1/2; z2 ((k − a + c − n)/2) a b (k + c − a − n)z (a + c − n)(k − a + c − n)   (2, a + 1), (2, b + 1), (k + a + c − n)/2 + 1 × 5 F4 (2, a + c − n + 1), (k − a + c − n)/2 + 1, 3/2; z2

+

((k + a + c − n + 1)/2) ((k − a + c − n + 1)/2)    (2, a ), (2, b ), (k + a + c − n + 1)/2 × 5 F4 (2, a + c − n), (k − a + c − n + 1)/2, 1/2; z2   a b z (2, a + 1), (2, b + 1), (k + a + c − n + 1)/2 , + 5 F4 (2, a + c − n + 1), (k − a + c − n + 1)/2, 3/2; z2 a+c−n

+

(3.2)

where (2, a) = a/2, (a + 1)/2. (b) Alternative expression can be derived in the following way. By virtue of the known transformation 2 F1 (a, b; c; w)

= (1 − w)c−a−b 2 F1 (c − a, c − b; c; w),

and posing w = 1/2 we obtain the expansion     ∞  (a )k (b )k  z k 1 c − a + k, c − b + k F3 a, a , b, b ; c; , z = 2a+b−c F 2 1 c + k; 1/2 2 (c)k k! 2 k=0

6

YU. A. BRYCHKOV

from the formula (3.1). Now we can make use of the relation  √ π( a+b+n a+b+n 1 2 ) ; = F a, b; 2 1 b−a−n 2 2 ( 2 )n   n    n 1 1 −k × (−2) (a)k + a+k+1 b+k−n b+k−n+1 k ( )( ) ( a+k ) 2 2 2 )( 2 k=0 

([10, 8.1.1.130], where 1 in the last term should be replaced be 1/2), with a → a + k, b → b + k, and obtain, after summation and simplification, the relation    n (−1)n 2n−a−1 (c)  1 k n (−1) F3 a, a , n − a, b; c; , z = k 2 (c − a)(1 − a)n k=0     ((k − a + c)/2) (2, a ), (2, b), (k − a + c)/2 × 5 F4 (2, c − a), (k + a + c)/2 − n, 1/2; z2 ((k + a + c)/2 − n)   a b(k − a + c)z (2, a + 1), (2, b + 1), (k − a + c)/2 + 1 − 5 F4 (2, c − a + 1), (k + a + c)/2 − n + 1, 3/2; z2 (a − c)(k + a + c − 2n)    ) ( k−a+c+1 (2, a ), (2, b), (k − a + c + 1)/2 2 + F 5 4 (2, c − a), (k + a + c + 1)/2 − n, 1/2; z2 ( k+a+c+1 − n) 2   a bz (2, a + 1), (2, b + 1), (k − a + c + 1)/2 − . (3.3) 5 F4 (2, c − a + 1), (k + a + c + 1)/2 − n, 3/2; z2 a−c 



(c) From the equality   n 1 (b)  1 (k+a,k−b−n) −b √ = 2 P F a, −n − a; b; π (0) 2 1 2 (a + b)n k! n−k k=0   1 1 + × b−a−k+1 ( a+b−k ) ( a+b−k+1 )( b−a−k 2 )( 2 2 2 ) [10, 8.1.1.132], we obtain n    2c−2 (c)(a + 1)n 1 n −2k 2 F3 (a, a , −n − a, b ; c; , z) = √ k 2 π(c − a)(a + c + n + 1) 



n−k 





k=0

n − k (k + a − c)m (−2)−m m (a + 1)k+m m=0       a+c−k (a + c + n)(k + a − c + m) c−a−k+1  +1 × 2 2 2 a−c × (a − c + 1)k (−a − c)k

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS



7



 (2, a ), (2, b ), (2, a + c), (c − a − k + 1)/2 × 7 F6 (2, a + c + n), (2, c − a − k − m), (a + c − k + 1)/2, 1/2; z2   (2, a + 1), (2, b + 1), (2, a + c + 1), (c − a − k + 1)/2; z2 − a b z7 F6 (2, a + c + n + 1), (2, c − a − k − m + 1), (a + c − k + 1)/2, 1/2      a+c−k+1 (a + c + n)(a + c − k)(a − c + k + m) c−a−k  + 2 2 (a + c)(a − c + k)   (2, a ), (2, b ), (2, a + c), (c − a − k)/2 + a bz + 7 F6 (2, a + c + n), (2, c − a − k − m), (a + c − k)/2, 1/2; z2 ⎛ ⎞⎤⎫ c−a−k+2 ⎪   ⎬ (2, a + 1), (2, b + 1), (2, a + c + 1), ⎜ ⎟⎥ 2 ×7 F 6 ⎝ . ⎠ ⎦ 3 a+c−k+2 ⎪ ⎭ + 1, ; z2 (2, a + c + n + 1), (2, c − a − k − m + 1), 2 2 (3.4) (d) Setting a = b = a = b = r = 1, c = n+2, in the integral representation F3 (a, a , b, b ; c + r; w, z)  1 1 = t c−1 (1 − t)r−1 2 F1 (a, b; c; wt)2 F1 (a , b ; r; z(1 − t)) dt B(c, r) 0

(3.5)

[11,(14)], where Rec, Rer > 0 and B(c, r) = (c)(r)/ (c + r) is the beta function, and using the formula   k  n  z 1 n+1 −n−1 n (n + 1)z (1 − z) ln(1 − z) − 2 F1 (1, 1; n + 2; z) = (−1) k z−1 k=1

[10, 8.1.1.12], we obtain after integration ⎧ (−w)−n−1 ⎨ (1 − w)n+1 F3 (1, 1, 1, 1; n + 3; w, z) = (n + 1)(n + 2) w + z − wz ⎩      z − wz z − wz , 2, n + 1 − ln(1 − w) , 1, n + 1 ×  w + z − wz w + z − wz

  n z w + z − wz  (−w)k − , 1, n + 1 + w + z + wz z−1 k! ×

p=0



k=1

 z n−k p (−w)  , 1, k + p + 1 , p z−1

n  



(3.6)

where (z, s, a) is the Lerch function. This formula can be also written in terms of the polylogarithm Lis (z) using the relation   n−1 k  z −n Lis (z) − . (z, s, n) = z ks k=1

8

YU. A. BRYCHKOV

We have, as well, F3 (1, 1, 1, 1; 2; w, z) =

ln[(1 − w)(1 − z)] , wz − w − z

(3.7)

see [12, 7.2.4.76] with a = b = 1, c = 2. (e) One can represent the function F3 (1, 1/2, 1, 1; 7/2; w, z) through logarithms and polylogarithms. However, the expression is very cumbersome. In this connection, we give only the result for w = z = 1/2 as an example: √   √ 7 1 1 45 2π 1 F3 1, , 1, 1; ; , = 45 − √ − π arctan 2 2 2 2 2 3 2 2   √  √  √ √ 1− 2 1− 2 2 1+ 2 1+ 2 +i + Li2 −i + arcsinh 1 + Li2 3 3 3 3 3  

  √ √ √ √ 1+ 2 1+ 2 1− 2 1− 2 −Li2 +i − Li2 −i . (3.8) 3 3 3 3 (f) Substituting a = 1, b = 2, c = 3/2, a = 1/2, b = 1/2, r = 1 into (3.5) and evaluating the integral we obtain    3 w + z − wz 1 1 5 F3 , 1, , 2; ; w, z = √ arctan 2 2 2 (1 − w)(1 − z) 2z (1 − z)(w + z − wz) √ 3 − √ arcsin w. (3.9) 2 wz (g) The formula    1 − 2z b z F3 a, b, 1 − a, 1 − b; 1 − b; z, = 2z − 1 1−z   a/2, (a + 1)/2 × 2 F1 , |z|, |z/(2z − 1)| < 1; b = 1, 2, . . . 1 − b; 4z − 4z2 

was derived in [13, (3.1)].

4. Reduction formulas for 2 (a) Using the formulas of confluence for F3 (a, a , b, b ; c; w, z), we can obtain from (3.5) an integral representation for the Humbert function 2 : 1 2 (b, b ; c + r; w, z) = B(c, r) 



1 0

t c−1 (1 − t)r−1 1 F1 (b; c; wt)1 F1 (b ; r; z(1 − t)) dt, (4.1)

Rec, Rer > 0. For b = r = 1, b = c/2, w = z, we get       c c + 1 (1−c)/2 z 1 (c−1)/2 −zt/2 zt c−1 z dt, e t e I(c−1)/2 2 , 1; c + 1; z, z = 2 c 2 2 2 0

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

9

and, evaluating the integral, we obtain   z  z & 1 1/2−b z/2 % 2b−1 z + Ib+1/2 . (4.2) 2 (b, 1; 2b + 1; z, z) = 2  b+ e Ib−1/2 2 2 2 If we put z → z/2, b = r = 1, b = c/2, w = z in (4.1), we obtain after integration    z2 z 1 z/2 = e 1 F2 b; b + , b + 1; . 2 b, 1; 2b + 1; z, 2 2 16 In particular,   z% z π  z & π z/2  z   z  1 2 , 1; 2; z, = ez/2 I0 1 + L1 − e I1 L0 , 2 2 2 2 2 2 2 2

(4.3)

(4.4)

Lν (z) is the modified Struve function. The formula (4.3) can be generalized. For this, we use the integral representation 2 (b, b ; c; w, z) =

ez B(b , c − b )



1 0





xc−b −1 (1 − x)b −1 e−zx 1 F1 (b; c − b ; zx) dx, (4.5)

Reb , Re(c − b ) > 0, which can be obtained from (4.1) with r = b . After substitution b = m + 1, c → c + m + 1, we obtain  1 ez 2 (b, m + 1; c + m + 1; w, z) = xc−1 (1 − x)m e−zx 1 F1 (b; c; wx) dx. B(m + 1, c) 0 (4.6) Now we substitute b → b + 1/2, c → 2b + 1 and get   (2b + m + 2) 1 2 b + , m + 1; 2b + m + 2; w, z = 2 (m + 1)(2b + 1)    1 1 2b m (1−x)z x (1 − x) e × 1 F1 b + ; 2b + 1; wx dx 2 0     wx  1 (b + 1) w −b z = dx, e xb (1 − x)m e(w/2−z)x Ib B(m + 1, 2b + 1) 4 2 0 since

  wx −b  wx  1 . b + ; 2b + 1; wx = (b + 1) ewx/2 Ib 2 4 2

 1 F1

If m is positive integer or zero, we can expand (1 − x)m and, for w = 2z, after evaluation of the integrals and some simplifications, we obtain the relation 2

  m−1 (2b)m z/2  (−1)k m − 1 z = e b, m; 2b + m; z, k 2 (m − 1)! 2b + k k=0   b + k/2; z2 /16 × 1 F2 , b + 1/2, b + k/2 + 1



(4.7)

10

YU. A. BRYCHKOV

m ≥ 1. The next step of generalization is based on the differentiation formula Dnz [zc−1 2 (b, b ; c; z, wz)] = (−1)n (1 − c)n zc−n−1 2 (b, b ; c − n; z, wz)

(4.8)

[8,(4.15)]. We have %

Dnz

z

×

2b+m−1

2

  m−1 (2b)m  (−1)k m − 1 z & = b, m; 2b + m; z, k 2 (m − 1)! 2b + k



k=0

    n b + k/2; z2 /16 n−p p Dz [z2b+m−1 ez/2 ]Dz 1 F2 . k b + 1/2, b + k/2 + 1

n   p=0

Now we can make use of the formulas Dnz [zλ e−az ] = n!zλ−n e−az Lλ−n n (az), where Lλn (z) is the Laguerre polynomial, and ⎛ 1 2⎞   ' (a ;z ) + n, n + p (ap )n (ap ⎜ 2 ⎟ 2n 2n 1 ' =2 Dz p Fq ⎠, p+1 Fq+1 ⎝ (bq ) 1 2 n (bq )n (bq ) + n, 2 ⎛ 3 2⎞ '      (a ;z ) + n + 1, n + p (ap )n+1 3 (ap ); z2 ⎜ 2 ⎟ 2n+1 ' = 2 D2n+1 F z F ⎠ p+1 q+1 ⎝ p q z (bq ) 3 2 n (bq )n+1 (bq ) + n + 1, 2 



); z2



[10,1.30.1.4-5], and after some transformation we obtain, for integer m ≥ 1 and n, the representation   m−1  (−1)n (2b)m ez/2 z k m−1 = 2 b, m; 2b + m − n; z, (−1) k 2 (m − 1)!(1 − 2b − m)n k=0 ⎡ 2b+m−n+2p−1 [n/2]  (z/4)2p Ln−2p (−z/2) ×⎣ p!(b + 1/2)p 2b + k + 2p p=0   b + k/2 + p, p + 1/2; z2 /16 × 2 F3 b + p + 1/2, b + k/2 + p + 1, 1/2 

[(n−1)/2] 

2b+m−n+2p

Ln−2p−1 (−z/2) (z/4)2p p!(b + 1/2)p+1 2b + k + 2p + 2 p=0   b + k/2 + p + 1, p + 3/2; z2 /16 . ×2 F3 b + p + 3/2, b + k/2 + p + 2, 3/2

z2 + 8

(b) Starting from the known formula 2 (b, b ; b + b ; w, z) = ez 1 F1 (b; b + b ; w − z)

(4.9)

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

11

and using the differentiation formulas (4.8) and 



Dnz [zb +n−1 2 (b, b ; c; w, z)] = (b )n zb −1 2 (b, b + n; c; w, z)

(4.10)

[8,(4.13)] for both variables, we obtain, after transformations and simplifications, the representation n    zk m!(−w)p ez n 2 (b + m, b + n; b + b − p; w, z) = k (b + b )k (b)m (1 − b − b )p k=0  m p    (b + k)r  p r (1 − b − b − k − r)s × (−w)  s r!(b + b + k) r r=0 s=0 



b+p+r−s−1

× (−w)−s Lm−r

(−w)1 F1 (b − s; b + b + k + r − s; w − z)]

(4.11)

for any non-negative integers m, n, p. If we pose b = r = 1, b = n, c = 2n in (4.5), we obtain 2 (n, 1; n + 1; w, z) = 2n e

z



1 0

t 2n−1 1 F1 (n; 2n; wt) e−zt dt,

and, due to the equality   z 1  z 1/2−a z/2 , F (a; 2a; z) =  a + e Ia−1/2 1 1 2 4 2 we get      1 1 wt w1/2−n ez dt. 2 (n, 1; 2n + 1; w, z) = 22n n n + t n−1/2 e(w−2z)t In−1/2 2 2 0 (4.12) Substituting the sum representation of the modified Bessel function of the half-integer argument (see, e.g. [14]), we obtain the formulas

2 (2n, 1; 4n + 1; w, z) = 2

4n+1

n   (1/2)2n −4n z  2n − 1 w n e 2k − 1 (2n − 1)! k=1

2k

× (4n − 2k − 1)!w [z

−2k

γ (2k, z) + (z − w)

−2k

γ (2k, z − w)] +

 n   2n − 1 k=1

2k − 2

× (4n − 2k)!w2k−1 [z1−2k γ (2k − 1, z) − (z − w)1−2k γ (2k − 1, z − w)]

(4.13)

12

YU. A. BRYCHKOV

for n = 1, 2, . . ., and 2 (2n + 1, 1; 4n + 3; w, z) = 2

4n+2

n+1   (1/2)2n+1 −4n−2 z  2n w (2n + 1) e 2k − 1 (2n)! k=1

2k

× (4n − 2k + 1)!w [z

−2k

γ (2k, z) + (z − w)

−2k

 n+1   2n γ (2k, z − w)] + 2k − 2 k=1

× (4n − 2k + 2)!w2k−1 [z1−2k γ (2k − 1, z) − (z − w)1−2k γ (2k − 1, z − w)] , (4.14) n = 0, 1, 2, . . .

5. Reduction formulas for 3 (a) From (4.1), due to the confluence formula, we obtain an integral representation for the Humbert function 3 :  1 1 (1 − t)c−1 t r−1 1 F1 (b; c; w(1 − t))0 F1 (; r; zt) dt. (5.1) 3 (b; c + r; w, z) = B(c, r) 0 Substituting b = c = 1 and simplifying the integrand yields  1 √ −c/2 w 3 (1; c + 2; w, z) = z e (c + 2) t c/2 e−wt Ic (2 zt) dt. 0

For c = −1/2 we get the following equality after evaluation of the integral: √  √   √ w+z/w     πe 3 w− z w+ z erf + erf , 3 1; ; w, z = √ √ √ 2 4 w w w

(5.2)

where erf(z) is the error function. The formula of differentiation of 3 Dnz [zc−1 3 (b; c; wz, z)] = (−1)n (1 − c)n 3 (b; c − n; wz, z) [8,(4.16)]] with b = 1, c = 3/2, yields after some evaluations √   (−1)n πwn−1/2 3 3 1; − n; w, z = 2 4(−1/2)n       z √ z √ w+z/w + w − erf − w × e erf w w ⎫     k−1 n ⎬  √ n k−1 w−p zp/2 I−p−1/2 (2 z) . (−1)k (−1)p −2w−1/2 z1/4 k p ⎭ k=1

(5.3)

p=0

Another differentiation formula Dnz [zb+n−1 3 (b; c; w, z)] = (b)n zb−1 3 (b + n; c; w, z)

(5.4)

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

13

[15,(75)] allows us to generalize (5.3) to the representation √   (−1)n π 3 3 m + 1; − n; w, z = 2 4(−1/2)n       z √ z √ n−1/2 w+z/w + w − erf − w × w e erf w w m  z  2wn−1 z1/4  n+p+1/2 − (−1)p Lm−p (−w)Lp − × w m! p=0 ×

n  k=1

(−1)k

  k−1   √ n k − 1p (n − p)m (−w)−p zp/2 I−p−1/2 (2 z) k p=0

 m m−k k−1   z  2wn−1  (−w)k  k−1 p k+n+p+1/2 − √ (−1) Lm−k−p (−w)Lp − r k! p=0 w r=0 π k=1 √  √   √  √ z z sinh(2 z) × (−4w)−r cosh(2 z)H2r √ + H2r+1 √ . √ w 2 w w

(5.5)

(b) Substituting (5.2) into the formula Dnz [3 (b; c; w, z)] =

1 3 (b; c + n; w, z) (c)n

(5.6)

[15,(62)] yields, after some transformations, the representation     3 3 3 1; n + ; w, z = w−n  n + 2 2       −1/2   √ √ z z w w+z/w e + erf × erf w+ w− 2 w w ⎫     n k−1  √ ⎬ n k−1 wp z−p/2 Ip+1/2 (2 z) . +z−1/4 (−1)k (−1)p k p ⎭ k=1

p=0

Applying (5.4) again, we obtain  √   π 3 3 3 m + 1; n + ; w, z = 2 2 2 n       z √ z √ (−1)m −n−1/2 w+z/w w + w + erf − w × e erf 2 w w m  (−w)k n−m−1/2  z  w−n z−1/4 Lm−k + − × k! w m! 

k=0

(5.7)

14

YU. A. BRYCHKOV

×

n  k=1

  k−1   √ n k − 1p (p − n + 1)m wp z−p/2 Ip+1/2 (2 z) (−1) k k

p=0

 m m−k k−1   z  wm−n−1  (−1)k  (−w)−p k−1 m−n−1/2 Lp − √ − r k! (m − k − p)! w π p=0 r=0 k=1 √  √   √  √ z z sinh(2 z) −r × (−4w) cosh(2 z)H2r √ + H2r+1 √ . √ w 2 w w

(5.8)

6. Reduction formulas for 1 , 2 Some relations for the confluent Appell functions (Humbert functions) 1 , 2 can be derived from formulas (3.1)–(3.3) by the confluence formulas  1 lim F3 a, a , b, ; c; w, εz = 1 (a, a , b; c; w, z), (6.1) ε→0 ε   1 1 2 lim F3 a, , b, ; c; w, ε z = 2 (a, b; c; w, z) : (6.2) ε→0 ε ε     n (−1)n 2n−a−1 (c)  1  k n 1 a, a , n − a; c; , z = (−1) k 2 (c − a)(1 − a)n k=0     ((k − a + c)/2) (2, a ), (k − a + c)/2; z2 /4 × 3 F4 (2, c − a), (k + a + c)/2 − n, 1/2 ((k + a + c)/2 − n)    a (k − a + c)z (2, a + 1), (k − a + c)/2 + 1; z2 /4 − 3 F4 (2, c − a + 1), (k + a + c)/2 − n + 1, 3/2 (a − c)(k + a + c − 2n)    ((k − a + c + 1)/2) (2, a ), (k − a + c + 1)/2; z2 /4 + 3 F4 (2, c − a), (k + a + c + 1)/2 − n, 1/2 ((k + a + c + 1)/2 − n)     az (2, a + 1), (k − a + c + 1)/2; z2 /4 , (6.3) − 3 F4 (2, c − a + 1), (k + a + c + 1)/2 − n, 3/2 a−c     n (−1)n 2n−a−1 (c)  1 k n 2 a, n − a; c; , z = (−1) k 2 (c − a)(1 − a)n k=0     ((k − a + c)/2) (k − a + c)/2; z2 /16 × 1 F4 (2, c − a), (k + a + c)/2 − n, 1/2 ((k + a + c)/2 − n)   (k − a + c)z (k − a + c)/2 + 1; z2 /16 − 1 F4 (2, c − a + 1), (k + a + c)/2 − n + 1, 3/2 (a − c)(k + a + c − 2n)    ((k − a + c + 1)/2) (k − a + c + 1)/2; z2 /16 + 1 F4 (2, c − a), (k + a + c + 1)/2 − n, 1/2 ((k + a + c + 1)/2 − n)   z (k − a + c + 1)/2; z2 /16 − , (6.4) 1 F4 (2, c − a + 1), (k + a + c + 1)/2 − n, 3/2 a−c 



INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

15

whence we have, for example, √   π 1 1 3 1 1 − , , ; 1; , z = √ 2 2 2 2 2 z % z z z z  z & 2 + zI−1/4 I3/4 + zI1/4 I5/4 , × I1/4 2 2 2 2 2   1 1 3 1 2 − , ; 1; , z = √ 2 2 2 2 *π % √ √ √ √ & 2 ber0 ( 2z) + ber21 ( 2z) + bei20 ( 2z) + bei21 ( 2z) × 2 √ √ √ √ + ber0 ( 2z)kei0 ( 2z) + ber1 ( 2z)kei1 ( 2z) √ √ √ √ + − bei0 ( 2z) ker 0 ( 2z) − bei1 ( 2z) ker 1 ( 2z)

(6.5)

(6.6)

for a = −1/2, a = 1/2, c = 1, n = 1 for (6.5) and n = 0 for (6.6); Iν (z) is the modified Bessel function of the first kind, berν (z), beiν (z), ker ν (z), keiν (z) are the Kelvin functions. (b) The representation 2 (a, b; c; w, z) =  ×

1 0

(s) (1−t)/2 z (s − t)

√ xt−1 (1 − x)c−t−1 2 F1 (a, b; c − t; (1 − x)w)It−1 (2 xz) dx.

Ret, Re(c − t) > 0 and |w| < 1 [15,(51)] implies the relation √   1 cosh(2x z) 3 1, 1; ; w, z = dx, 2 2 0 1 − w + wx

 2

from which, after evaluation of the integral, we have     1 w − 1√ 3 2 1, 1; ; w, z = √ √ exp 2 z 2 w w w−1        √ √ w − 1√ w − 1√ × Ei 2 z − 2 z − Ei −2 z − 2 z w w       √ w − 1√ w − 1√ + exp −2 z Ei −2 z + 2 z w w  

 √ w − 1√ −Ei 2 z + 2 z w 

for 0 ≤ arg w ≤ 3π/4, 0 ≤ arg z ≤ π; Ei(z) is the integral exponential function.

(6.7)

16

YU. A. BRYCHKOV

(c) One more reduction formula has the form    n   1  n 3 w k (b)k √ −n, b; ; w, −z = √ k 2 2 z z

 2

k=0

  √ −m √ (k + m)! k−m (−4 z) sin 2 z + π . × m!(k − m)! 2 m=0 k 

(6.8)

7. Generating functions Substituting the values a = −n, b = −m (m, n are non-negative integers) into the series expansion (3.1) we obtain the relation n   (b)k m!  n (c+k−1,b −c−k−m) Pm (1 − 2z) F3 (−n, −m, b, b ; c; w, z) = k (c)n (c + m)k 

(7.1)

k=0

or, equivalently   (c)n 1−z  F3 −n, −m, b, b + c + m + 1; c + 1; −w, m! 2   n  n (b)k (c+k,b −k) = wk Pm (z); k (c + m)k

(7.2)

k=0

the last formula represents a finite generating function for the Jacobi polynomials. Analogous series expansions for the functions 1 and 2 reduces to the generating functions 1 (−n, 1, b; c + 1; w, z) = cz

n    n (b)k  w k − e γ (c + k, z) k (c)k z

−c z

(7.3)

k=0

for the complementary incomplete gamma-function γ (ν, z), n   (b)k m!  n 1 (−n, −m, b; c; w, z) = (−w)k Lc+k−1 (z) m k (c)m (c + m)k

(7.4)

k=0

for the Laguerre polynomials Lλn (z), 2 (−n, b; c; w, z) = (c)z

(1−c)/2

n    n k=0

k

√ (b)k (−w)k z−k/2 Ic+k−1 (2 z)

for the modified Bessel function Iν (z).

Disclosure statement No potential conflict of interest was reported by the author.

(7.5)

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

17

References [1] Bytev VV, Kalmykov MYu, Moch S-O. HYPERDIRE: HYPERgeometricfunctions DIfferential REduction MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: FD and FS horn-type hypergeometric functions of three variables. Comput Phys Commun. 2014;185:3041–3058. [2] Kniehl BA, Tarasov OV. Finding new relationships between hypergeometric functions by evaluating Feynman integrals. Nucl Phys B. 2012;854(3):841–852. [3] Brychkov YuA. On some properties of the Nuttall function Qν (a, b). Integral Transforms Spec Funct. 2014;25(1):33–43. [4] Sofotasios PC, Tsiftsis TA, Brychkov YuA, et al. Analytic expressions and bounds for special functions and applications in communication theory. IEEE Trans Inform Theory. 2014;60(12):7798–7823. [5] Brychkov YuA, Savischenko NV. A special function of communication theory. Integral Transforms Spec Funct. 2015;26(6):470–484. [6] Brychkov YuA, Savischenko NV. Some properties of the Owen T-function. Integral Transforms Spec Funct. 2016;27(2):163–180. [7] Appell P, Kampé de Fériet J. Fonctions hypergéométriques et hypersphériques: polynomes d’Hermite. Paris: Gauthier-Villars; 1926. [8] Brychkov YuA, Saad N. On some formulas for the Appell function F1 (a, b, b ; c; w; z). Integral Transforms Spec Funct. 2012;23(11):793–802. [9] Shpot MA. A massive Feynman integral and some reduction relations for Appell functions. J Math Phys. 2007;48:123512. [10] Brychkov YuA. Handbook of special functions: derivatives, integrals, series and other formulas. Boca Raton (FL): Chapman & Hall/CRC; 2008. [11] Luke YL. Inequalities for generalized hypergeometric functions of two variables. J Approx Theory. 1974;11:73–84. [12] Prudnikov AP, Brychkov YuA, Marichev OI. Integrals and series. More special functions. Vol. 3. New York (NY): Gordon and Breach; 1990. [13] Vidunas R. Dihedral Gauss hypergeometric functions. Kiushu J Math. 2011;65:141–167. [14] http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/03/01/02/02/. [15] Brychkov YuA, Saad N. On some formulas for the Appell function F3 (a, b, b ; c, c ; w; z). Integral Transforms Spec Funct. 2015;26(11):910–923.