paper are (pxp) real symmetric positive definite matrices and the meanings of all the other symbols used are the same as in the works of Mathai [5, 6]. 1.
ON EXTON’S GENERALIZED QUADRUPLE HYPERGEOMETRIC FUNCTIONS AND CHANDEL’S FUNCTION OF MATRIX ARGUMENTS Lalit Mohan Upadhyaya* and H.S. Dhami**, Department of Mathematics, University of Kumaun, Almora Campus, Almora, Uttaranchal, India- 263601. 2000 AMS Mathematics Subject Classification: Primary: 33C65, 33C99. Secondary: 60E, 62H, 44A05. Key Words: Chandel’s
(k) (n) function, Exton’s (k) (n) and E E (1) C (1) D
(k) (n) functions, matrix arguments, matrix transform. E (2) D ABSTRACT In continuation of our previous studies [6,7] we have established five results in this
(k) (n) (k) (n) E function, one for the Exton’s E (1) C (1) D (k) (n) E function of matrix arguments. function and three for the Exton’s (2) D paper- one for the Chandel’s
INTRODUCTION Exton [2,3] has given two functions
(k) (n) and (k) (n) which, according to him, E E (1) D (2) D
are generalizations of certain of the quadruple hypergeometric functions discussed by him in [3].Chandel [1] has also given a similar function already defined the functions
(k) (n) . We have E (1) C
(k) (n) and (k) (n) for the matrix arguments case E E (1) D (1) C
in our previous article [7]. * Department of Mathematics, Municipal Post Graduate College, Mussoorie, Dehradun, Uttaranchal, India-248179. ** To whom all the correspondence may be addressed.
1
In this paper we have defined the function
(k) (n) E with matrix arguments and (2) D
have proved some results for these functions. All the matrices appearing in this paper are (pxp) real symmetric positive definite matrices and the meanings of all the other symbols used are the same as in the works of Mathai [5, 6]. 1. The Chandel’s
(k) (n) Function E (1) C
THEOREM 1.1:
(k) (n) ' E (a, a , b; c , , c ; − X , , − X ) (1) C 1 n 1 n 1 e− tr(U + V + W) U a − (p +1) / 2 × = ∫ ∫ ∫ U 0 V 0 W 0 > > > ' Γp (a)Γp (a ) Γp (b) ' 1 1 1 1 V a − (p +1) / 2 W b − (p +1) / 2 F ( ; c ; − W 2 U 2 X U 2 W 2 ) × 0 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 F ( ; c ; −W U X U W ) F ( ; c ; W V X V W 2) × 0 1 k k 0 1 k +1 − k +1 1 1 1 1 2 2 2 0 F1 ( ; cn ; − W V X V W 2 )dUdVdW (1.1) n ' for Re(a,a , b) > (p − 1) / 2. PROOF: Taking the M-transform (matrix transform) of the right side of eq.(1.1) with respect to the variables X , , X and the parameters ρ , , ρ respectively,
1
n
1
n
we have,
ρ − (p+1) / 2 ρ − (p+1) / 2 ρ − (p +1) / 2 + 1 k k 1 × X X X ∫X >0 ∫X > 0 1 k k +1 1 n 1 1 1 1 ρ − (p +1) / 2 2 2 2 − X n F ( ; c ; W U X U W 2 ) × n 0 1 1 1 1 1 1 1 1 1 1 1 F ( ; c ; −W 2 U 2 X U 2 W 2 ) F ( ; c ; −W 2 V 2 X V 2W 2) × 0 1 k k 0 1 k +1 k +1 Continued to the next page ……………………………..
2
1
1 1 1 2 2 2 0 F1( ; c n ; − W V X n V W 2 )dX1 dX k dX k 1 dX n +
(1.2)
Applying the following transformations to the expression (1.2),
1 1 1 1 1 1 1 1 Y = W 2 U 2 X U 2 W 2 , Y = W 2 V 2 X V 2 W 2 with dY = W (p +1) / 2 × i i j j i U (p+1) / 2 dX ; dY = W (p+1) / 2 V (p+1) / 2 dX ; and Y = W U X ; i j j i i Y = W V X ; for i = 1, , k and j = k + 1, , n j j and then writing the M-transforms of the F functions, we are led to, 0 1 Γp (c1) Γp (ρ1 ) −ρ − −ρ −ρ − −ρ −ρ − −ρ U 1 × k V k +1 n W 1 n Γp (c1 − ρ1) Γ p (c n )Γ p (ρ n ) (1.3) Γ p (c n − ρ n ) On substituting this expression on the right side of eq.(1.1) and then integrating out the variables U,V,W in the resulting expression by using a Gamma integral produces
M[
(k) (n) as given by eq.(3.2) of the authors’ paper [7]. E ] (1) C 2. The Exton’s
(k) (n) Function E (1) D
THEOREM 2.1:
(k) (n) ' E (a, b , , b ; c, c ; − X , , − X ) (1) D 1 n 1 n =
− tr(U1 ++ U n ) b − (p +1) / 2 1 1 e U × ∫ ∫ U 0 U 0 1 > > Γp (b1) Γp (b n ) 1 n
1 1 1 1 1 1 b − (p +1) / 2 ' 2 2 2 2 2 n U Ψ 2 (a; c, c ; − U1 X1U1 − − U k X k U k , − U k 1X k 1U k 21 n + + + 1 1 2 − − U n X n U n 2 )dU1 dU n (2.1) Continued in the next page ……………………………..
3
for Re(b , , b ) > (p − 1) / 2. 1 n PROOF: Taking the M-transform of the right side of eq.(2.1) with respect to the variables X , , X and the parameters ρ , , ρ respectively, we get,
1
n
1
n ρ k − (p+1) / 2
ρ1 − (p+1) / 2 ρ k 1 − (p+1) / 2 + X X X × ∫X > 0 ∫X > 0 1 k k +1 1 n 1 1 1 1 ρn − (p +1) / 2 ' 2 2 2 Xn Ψ 2 (a; c,c ; −U1 X1U1 − − U k X k U k 2 , 1 1 1 1 2 (2.2) − U k 1X k 1U k 21 − − U n 2 X n U n 2 )dX1 dX k dX k 1 dX n + + + + Applying the transformations
1 (p +1) / 2 2 2 Y =U X U with dY = U dX and Y = U X for j = 1, , n j j j j j j j j j j 1
to the above expression generates, −ρn ρ1 −(p +1) / 2 ρk −(p +1) / 2 −ρ1 U U Y Y × ∫Y >0 ∫Y > 0 1 1 n k 1 n ρ −(p +1) / 2 ρ − (p +1) / 2 '; Y Yk 1 k +1 (a; c, c Yn n Ψ − 1 − − Yk , 2 + (2.3) − Yk 1 − − Yn )dY1 dYk dYk 1 dYn + + Now, we apply the following transformations to the expression (2.3),
Z = Y , Z = Y + Y , , Z = Y + + Y ; Z , =Y 1 1 2 1 2 k 1 k k +1 k +1 Z Y Y , ,Z Y Y ; with dZ dZ = dY dY k + 2 = k +1 + k + 2 n = k +1 + + n 1 k 1 k and dZ dZ dY dY (from eq.(6.7) page 95 of Mathai [4]), where, k +1 n = k +1 n 0 < Z < < Z and 0 < Z Z . 1 k k +1 < < n Then integrating out the variables Z , , Z one-byand Z , ,Z 1 k −1 k +1 n −1 one and in order by using a type-1 Beta integral and on writing the M-transform of a Ψ - function, we obtain,
2
' −ρ1 −ρn Γp (ρ1) Γp (ρn ) Γp (c) Γp (c ) Γp (a − ρ1 − − ρn ) U Un (2.4) 1 ' Γ p (a)Γ p (c − ρ1 − − ρk )Γ p (c − ρ k 1 − − ρ n ) +
4
Substituting this expression on the right side of eq.(2.1) and then integrating out the variables U , , U in the resulting expression by using a Gamma integral, the
1
n (k) (n) as given by eq.(3.1) of the authors’ article [7]. outcome is M[ E ] (1) D 3. The Exton’s
(k) (n) Function E (2) D
(k) (n) - function of matrix arguments E (2) D (k) (n) (k) (n) ' E E (a, a , b , , b ; c; − X , , − X ) (2) D = (2) D 1 n 1 n
DEFINITION 3.1: The
is defined as that class of functions for which the matrix transform (M-transform) is the following:
ρ1 − (p+1) / 2 ρ n − (p+1) / 2 (k) (n) E ] = [∫ X X × ∫X > 0 1 (2) D X >0 n 1 n (k) (n) ' E (a, a , b , , b ; c; − X , , − X )dX dX ] (2) D 1 n 1 n 1 n ' Γp (a − ρ1 − − ρk ) Γp (a − ρk 1 − − ρn ) Γp (b1 − ρ1) Γp (b n − ρn ) + = × ' (b ) (b ) Γ p (a) Γ Γ Γ p (a ) p 1 p n M[
Γp (c) Γp (ρ1) Γp (ρn ) Γp (c − ρ1 − − ρ n )
(3.1)
' for Re(a − ρ − − ρ , a − ρ ,b ,c , ) 1 k k +1 − − ρn j − ρ j − ρ1 − − ρn ρ j > (p − 1) / 2; j = 1, , n. THEOREM 3.1:
(k) (n) E (a, a ' , b , , b ; c; − X , , − X ) (2) D 1 n 1 n Γ p (c) b − (p +1) / 2 = (n) ∫ U1 1 × ∫ Γp (b1) Γp (bn ) Γp (c − b1 − − b n ) Continued in the next page ………………….
5
U
b − (p +1) / 2 c − b −− b −(p +1) / 2 n 1 n I − U − − U × n 1 n 1 1 1 1 −a I + U 2X U 2 + + U 2X U 2 × 1 1 1 k k k
' a − 1 1 1 1 2 2 2 I+ U X U U X U 2 dU dU k +1 k +1 k +1 + + n n n 1 n
(3.2)
where U ' = U > 0 and 0 < U + + U < I and for Re(b , c − b − − b ) i i 1 n i 1 n > (p − 1) / 2; i = 1, , n. PROOF: Taking the M-transform of the right side of eq.(3.2) with respect to the variables X , , X and the parameters ρ , , ρ respectively, we have,
1
n
1
ρ1 − (p+1) / 2 X Xk ∫X >0 ∫X > 0 1 1 n Xn
ρn − (p +1) / 2
n ρk − (p +1) / 2
X
k +1
ρk 1 − (p +1) / 2 + ×
1 −a I + U 2X U 2 + + U 2 X U 2 × 1 1 1 k k k 1
1
1
' 1 −a I+ U 2 X U 2 U 2X U 2 dX dX dX dX (3.3) k +1 k +1 k +1 + + n n n 1 k k +1 n 1
1
1
On using the transformations
1 (p +1) / 2 2 2 Y = U X U , with dY = U dX ; and Y = U X for j = 1, , n j j j j j j j j j j in the above expression and then integrating out the variables Y , , Y and 1 k Y , , Y by using a type-2 Dirichlet integral, we are led to, k +1 n −ρ1 −ρn Γ p (ρ1) Γ p (ρn )Γ p (a − ρ1 − − ρk ) U1 Un × ' Γ p (a)Γ p (a ) ' (3.4) Γp (a − ρk 1 − − ρn ) + 1
6
Substituting this expression on the right side of eq.(3.2) and then integrating out U , , U in the resulting expression by using a type-1 Dirichlet integral we have
1
n (k) (n) M[ E ] as given by eq.(3.1). (2) D
It may be easily seen that a limiting form of eq.(3.2) has the following form:
− X1 −X n (k) (n) lim E ( , , b , , b ; c; , , ) (2) D α α 1 n α α α→∞ Γ p (c) b − (p +1) / 2 1 (n) U = × ∫ ∫ 1 Γp (b1) Γp (bn ) Γp (c − b1 − − b n ) b − (p +1) / 2 c − b −− b − (p +1) / 2 1 n U n I − U − − U × n 1 n − tr(U1X1 ++ U n X n ) e dU dU (3.5) 1 n where U ' = U > 0 and 0 < U + + U < I and for Re(b , c − b − − b ) i i 1 n i 1 n > (p − 1) / 2; i = 1, , n.
THEOREM 3.2: A case of reducibility:-
−X −X (k) (n) lim E (α, α, b , , b ; c; , (n) , ) (2) D 1 n α α α→∞ = 1F1(b1 + + b n ; c; − X)
PROOF: To prove this theorem we put X = X
1
2
(3.6)
= = X = X in eq.(3.5) and n
apply the following transformations to the resulting expression V = U , V = U + U , , V = U + + U ; with dU dU = dV dV 1 1 2 1 2 n 1 n 1 n 1 n (from eq.(6.7) page 95 of Mathai [4]), where,0 < V < < V < I. 1 n one-by-one and in order by Then integrating out the variables V , , V
1
n −1
using a type-1 Beta integral and finally using the theorem 2.3.4 page 42 of Mathai [5], the desired result follows.
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THEOREM 3.3:
(k) (n) E (a, a ' , b , , b ; c; −X , , − X ) (2) D 1 n 1 n tr(U + V) a − (p +1) / 2 a ' − (p +1) / 2 − e U V = × ' ∫U > 0 ∫V > 0 Γp (a)Γp (a ) 1 1 1 1 1 1 (n) 2 2 2 2 2 Φ 2 (b1, , b n ; c; − U X1U , , − U X k U , − V X k 1V 2 , , + 1 1 − V 2 X n V 2 )dUdV (3.7) ' for Re(a,a ) > (p − 1) / 2. 1
PROOF: Taking the M-transform of the right side of eq.(3.7) with respect to the variables X , , X and the parameters ρ , , ρ respectively, we get,
1
n
1
ρ1 − (p+1) / 2 X Xk ∫X >0 ∫X > 0 1 1 n
n ρk − (p+1) / 2
X
k +1
ρk 1 − (p +1) / 2 + ×
ρn − (p +1) / 2
1 1 1 1 (n) 2 2 2 Xn Φ 2 (b1, , b n ; c; −U X1U , , − U X k U 2 , 1 1 1 1 2 2 2 (3.8) − V X k 1V , , − V X n V 2 )dX1 dX kdX k 1 dX n + + Making use of the transformations
1
1 1 1 2 2 2 Y = U X U ; Y = V X V 2 with dY = U (p+1) / 2 dX ; i i j j i i dY = V j
(p+1) / 2
dX ; and Y = U X ; Y = V X for i = 1, , k; j = k + 1, , n j i i j j
to the expression (3.8) and then using eq.(1.4) of the authors’ paper [7], we obtain,
U −ρ1 −−ρ k V −ρk +1 −−ρ n
Γp (b1 − ρ1) Γp (ρ1) Γ p (b1)
Γ p (c) Γ p (c − ρ1 − − ρ n )
8
Γp (b n − ρn ) Γp (ρn ) × Γ p (bn ) (3.9)
Substituting the above expression on the right side of eq.(3.7) and then integrating out the variables U and V in the resulting expression by using a Gamma integral produces M[
(k) (n) E ] as given by eq.(3.1) above. (2) D
References 1. Chandel R.C. Singh (1973). On Some Multiple Hypergeometric Functions Related to Lauricella Functions; Jnanabha Sect. A 3, pp.119-136. 2. Exton H. (1972). On Two Multiple Hypergeometric Functions Related to
(n) ; Jnanabha Sect. A 2, pp.59-73. D
Lauricella’s F
3. Exton H. (1976). Multiple Hypergeometric Functions and Applications; Ellis Horwood Limited, Publishers, Chichester. 4. Mathai A.M. (1992). Jacobians of Matrix Transformations- I; Centre for Mathematical Sciences, Trivandrum , India. 5. Mathai A.M. (1993). Hypergeometric Functions of Several Matrix Arguments; Centre for Mathematical Sciences, Trivandrum , India. 6. Upadhyaya Lalit Mohan, Dhami H.S. (Nov.2001). Matrix Generalizations of Multiple Hypergeometric Functions; #1818 IMA Preprint Series, University of Minnesota, Minneapolis, U.S.A. 7. Upadhyaya Lalit Mohan, Dhami H.S. (Dec.2001). On Some Multiple Hypergeometric Functions of Several Matrix Arguments; #1821 IMA Preprint Series, University of Minnesota, Minneapolis, U.S.A.
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