appell's and humbert's functions of matrix arguments

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APPELL’S AND HUMBERT’S FUNCTIONS OF MATRIX ARGUMENTS – I Lalit Mohan Upadhyaya* & 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 : Appell’s functions, Humbert’s functions, matrix arguments, matrixtransform. ABSTRACT We have proved six results for the Appell’s functions of matrix arguments-two each for the functions F and F and one each for the functions F and F .

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INTRODUCTION Appell’s functions of matrix arguments have earlier been studied by Mathai [4, 5, 6] and also by Saxena, Sethi and Gupta [7]. In the present paper we have utilized Mathai’s definitions for all the functions studied. 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 [3,4]. 1. Preliminary Definitions

' DEFINITION 1.1: The Appell’s function F = F (a, b, b ; c;−X,−Y) of matrix 1

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arguments is defined as that function for which the M-transform (matrix-transform) is the following: * Department of Mathematics, Municipal Post Graduate College, Mussoorie, Dehradun (Uttaranchal), India, 248179. ** To whom all the correspondence may be addressed.

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M(F ) = ∫ X ρ1 − (p +1) / 2 Y ρ 2 − ( p +1) / 2 F (a , b, b' ; c;− X, − Y )dXdY ∫ 1 X >0 Y >0 1 Γp ( c)Γp (ρ1)Γp (ρ 2 )Γp (a − ρ1 − ρ 2 )Γp (b − ρ1 )Γp ( b ' − ρ 2 ) =  (1.1) ' Γp (a )Γp ( b)Γp ( b )Γp (c − ρ1 − ρ 2 ) for Re (a − ρ − ρ , b − ρ , b' − ρ , c − ρ − ρ , ρ , ρ ) > ( p − 1) / 2. 1 2 1 2 1 2 1 2 DEFINITION 1.2: F = F (a , b, b '; c, c '; − X,− Y) 2 2 ρ M (F ) = ∫ X 1 − (p +1) / 2 Y ρ 2 − (p +1) / 2 F (a , b, b'; c, c' ;−X,− Y)dXdY ∫ 2 2 X > 0 Y >0 Γ (c)Γ (c ' )Γ (ρ )Γ (ρ )Γ (a − ρ − ρ )Γ (b − ρ )Γ (b' − ρ ) p p p 1 p 2 p 1 2 p 1 p 2 =  (1.2) ' ' Γp (a )Γp (b)Γp (b )Γp (c − ρ1)Γp (c − ρ2 ) for Re (a − ρ1 − ρ 2 , b − ρ1, b' − ρ2 , c − ρ1, c' − ρ 2 , ρ1, ρ2 ) > (p − 1) / 2. DEFINITION 1.3: F = F (a , a ' , b, b '; c;− X ,− Y ) 3 3 M (F ) = ∫ X ρ1 −( p+1) / 2 Y ρ 2 − (p +1) / 2 F (a, a ' , b, b '; c; −X,−Y )dXdY ∫ 3 > > 3 X 0 Y 0 Γ (c)Γ (ρ )Γ (ρ )Γ (a − ρ )Γ (a ' − ρ )Γ ( b − ρ )Γ (b ' − ρ ) p p 1 p 2 p 1 p 2 p 1 p 2 =  (1.3) Γp (a )Γp (a ' )Γp (b)Γp (b' )Γp (c − ρ1 − ρ 2 ) for Re (a − ρ1, a ' − ρ 2 , b − ρ1, b ' − ρ 2 , c − ρ1 − ρ2 , ρ1, ρ 2 ) > (p − 1) / 2. DEFINITION 1.4: F = F (a , b; c, c '; − X,− Y ) 4 4 ρ M (F4 ) = ∫X > 0 ∫Y > 0 X 1 − (p +1) / 2 Y ρ 2 −( p +1) / 2 F4 (a , b; c, c' ;− X,− Y)dXdY Γp (c)Γp (c' )Γp (ρ1)Γp (ρ 2 )Γp (a − ρ1 − ρ 2 )Γp ( b − ρ1 − ρ2 ) =  (1.4) ' Γ (a )Γ (b)Γ (c − ρ )Γ (c − ρ ) p p p 1 p 2 for Re (a − ρ1 − ρ2 , b − ρ1 − ρ 2 , c − ρ1, c ' − ρ2 , ρ1, ρ2 ) > (p − 1) / 2.

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DEFINITION 1.5: The Humbert’s function Φ = Φ (a , b; c;− X ,− Y ) of matrix

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arguments is defined as that function which has the following M-transform:

M (Φ1) = ∫X 0 ∫Y 0 X ρ1 − (p +1) / 2 Y ρ 2 − (p +1) / 2 Φ1(a , b; c;− X,−Y )dXdY > > Γp (c)Γp (ρ1)Γp (ρ 2 )Γp (a − ρ1 − ρ2 )Γp ( b − ρ1) =  (1.5) Γp (a )Γp (b)Γp (c − ρ1 − ρ 2 ) for Re (a − ρ1 − ρ 2 , b − ρ1, c − ρ1 − ρ 2 , ρ1, ρ 2 ) > ( p − 1) / 2.

DEFINITION 1.6: Ψ = Ψ (a , b; c, c ' ;− X,− Y ) 1 1

M (Ψ1) = ∫X 0 ∫Y 0 X ρ1 − (p +1) / 2 Y ρ 2 − (p +1) / 2 Ψ1(a , b; c, c ';− X,−Y )dXdY > > Γp (c)Γp (c ' )Γp (ρ1)Γp (ρ 2 )Γp (a − ρ1 − ρ 2 )Γp (b − ρ1) =  (1.6) Γp (a )Γp (b)Γp (c − ρ1)Γp (c' − ρ2 )

for Re (a − ρ1 − ρ 2 , b − ρ1, c − ρ1, c ' − ρ 2 , ρ1, ρ 2 ) > (p − 1) / 2.

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DEFINITION 1.7: Ψ = Ψ (a; c, c ;−X,−Y) 2 2

M (Ψ2 ) = ∫X 0 ∫Y 0 X ρ1 −( p +1) / 2 Y ρ2 −( p+1) / 2 Ψ2 (a; c, c' ;−X,− Y)dXdY > > Γp (c)Γp (c ' )Γp (ρ1)Γp (ρ2 )Γp (a − ρ1 − ρ 2 ) =  (1.7) ' Γp (a )Γp (c − ρ1)Γp (c − ρ2 )

for Re (a − ρ1 − ρ 2 , c − ρ1, c ' − ρ 2 , ρ1, ρ 2 ) > (p − 1) / 2.

DEFINITION 1.8: Ξ = Ξ (a , b; c;− X,− Y )

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2 M (Ξ 2 ) = ∫X 0 ∫Y 0 X ρ1 − (p +1) / 2 Y ρ2 −( p+1) / 2 Ξ 2 (a , b; c;− X,− Y )dXdY > > Γp (c)Γp (ρ1)Γp (ρ 2 )Γp (a − ρ1)Γp (b − ρ1) =  (1.8) Γp (a )Γp (b)Γp (c − ρ1 − ρ 2 ) for Re (a − ρ1, b − ρ1, c − ρ1 − ρ 2 , ρ1, ρ 2 ) > (p − 1) / 2.

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2. Appell’s Functions of Matrix Arguments. THEOREM 2.1:

F (α, β, β'; γ;−X, −Y ) 1 Γp ( γ ) I I β + β' − (p +1) / 2 V β' − ( p +1) / 2 × U = ' ' ∫0 ∫0 Γp (β)Γp (β )Γp ( γ − β − β ) 1 1 1 1 ' − ( p +1) / 2 ( p 1 ) / 2 γ − β − β β − + 2 2 2 I−U I−V I + (I − V ) U XU ( I − V ) 2 1 −α dUdV + V 2 U 2 YU 2 V 2 1

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 (2.1)

for 0 < U < I, 0 < V < I and for Re (β, β ' , γ − β − β' ) > ( p − 1) / 2. PROOF: We take the M-transform of the right side of eq. (2.1) with respect to the variables X and Y and the parameters ρ and ρ respectively to obtain,

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1 1 1 1 ρ1 − (p +1) / 2 Y ρ 2 − (p +1) / 2 I + (I − V ) 2 U 2 XU 2 (I − V ) 2 X ∫X >0 ∫Y >0 1 −α dXdY + V 2 U 2 YU 2 V 2 1

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 ( 2.2)

Making use of the transformations,

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1 1 1 1 1 1 1 2 2 2 2 2 2 2 X1 = ( I − V ) U XU ( I − V ) , Y1 = V U YU V 2 with , dX = I − V ( p +1) / 2 U ( p +1) / 2 dX, dY = V (p +1) / 2 U (p +1) / 2 dY and 1 1 X1 = I − V U X , Y1 = V U Y in the above expression and then integrating out the variables X and Y by using a

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type-2 Dirichlet integral we get,

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Γp (ρ1)Γp (ρ 2 )Γp (α − ρ1 − ρ 2 ) − ρ − ρ − ρ − ρ U 1 2 V 2 I−V 1 Γp (α)

 (2.3)

Substituting this expression on the right side of eq. (2.1) and then integrating out the variables U and V in the resulting expression by using a type-1 Beta integral generates M ( F ) as given by eq. (1.1).

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THEOREM 2.2:

' − 12 − 1 2 − β ' P F (α, β, β ; γ ;− X,− P YP ) 1 1 1 ' − ( p +1) / 2 1 β tr ( PT ) − 2 T = Φ1(α, β; γ;− X ,− T YT 2 )dT ∫T > 0 e ' Γp (β ) for Re (β ' ) > ( p − 1) / 2.

 ( 2.4)

PROOF: Taking the M-transform of the right side of eq. (2.4) with respect to the variables X and Y and the parameters ρ and ρ respectively, we have,

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1 1 ( p 1 ) / 2 ( p 1 ) / 2 − + ρ − + ρ 2 Y 2 Φ1(α, β; γ; −X ,−T YT 2 )dXdY  (2.5) ∫X >0 ∫Y >0 X 1 which, under the transformation

1 1 Y1 = T 2 YT 2 ( with dY1 = T ( p +1) / 2 dY and Y1 = T Y ) and then using the definition (1.5) yields,

Γp (γ )Γp (ρ1)Γp (ρ 2 )Γp (α − ρ1 − ρ 2 )Γp (β − ρ1 ) − ρ T 2 Γp (α )Γp (β)Γp ( γ − ρ1 − ρ 2 )

 ( 2.6)

Substituting this expression on the right side of eq. (2.4) and then integrating out T in the resulting expression by using a Gamma integral gives,

Γp (β ' − ρ 2 )Γp ( γ )Γp (ρ1 )Γp (ρ 2 ) Γp (α − ρ1 − ρ 2 )Γp (β − ρ1) ' P − (β − ρ 2 )  ( 2.7) ' Γp (β )Γp (α)Γp (β)Γp ( γ − ρ1 − ρ 2 ) Now, taking the M-transform of the left side of eq. (2.4) with respect to the variables X and Y and the parameters ρ and ρ respectively, we get,

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' ( p 1 ) / 2 ( p 1 ) / 2 − + − β ρ − + ρ Y 2 P × ∫X >0 ∫Y >0 X 1 − 1 2 − 12 ' F (α, β, β ; γ; −X ,− P YP )dXdY 1

 (2.8)

which, under the transformation

Y2 = P

− 12

YP

− 12

(p 1) / 2 1 ( with dY2 = P − + dY and Y2 = P − Y )

and then using the definition (1.1) leads us to the same result as in eq. (2.7). THEOREM 2.3:

' − 12 − 1 2 − β ' ' P F ( α , β , β ; γ , γ ;− X , − P YP ) 2 1 1 ' − ( p +1) / 2 1 β tr ( PT ) ' − 2 T = Ψ1(α, β; γ , γ ;− X,− T YT 2 )dT ∫T > 0 e ' Γp (β ) for Re (β ' ) > ( p − 1) / 2.

 ( 2.9)

PROOF: Taking the M-transform of the right side of eq. (2.9) with respect to the variables X and Y and the parameters ρ and ρ respectively, we get,

1 2 ρ − ( p +1) / 2 Y ρ 2 − ( p +1) / 2 × ∫X > 0 ∫Y > 0 X 1 1 1 ' 2 Ψ1(α, β; γ , γ ;− X ,− T YT 2 ) dXdY

 (2.10)

The application of the same transformation to this expression as we have applied to the expression (2.5) above and then the use of definition (1.6) leads us to,

' Γp ( γ )Γp ( γ )Γp (ρ1 )Γp (ρ 2 )Γp (α − ρ1 − ρ 2 )Γp (β − ρ1 ) T −ρ2 Γp (α )Γp (β)Γp ( γ − ρ1)Γp ( γ ' − ρ 2 )

 (2.11)

Substituting this expression on the right side of eq. (2.9) and then integrating out T in the resulting expression by using a Gamma integral gives,

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Γ (β ' − ρ ) Γ ( γ ' )Γ ( γ) Γ (ρ )Γ (ρ )Γ (α − ρ − ρ )Γ (β − ρ ) ' p 2 p p p 1 p 2 p 1 2 p 1  ( 2.12) P − (β − ρ 2 ) ' ' Γp (β )Γp (α )Γp (β)Γp ( γ − ρ1) Γp ( γ − ρ 2 ) Now, taking the M-transform of the left side of eq. (2.9) with respect to the variables X and Y and the parameters ρ and ρ respectively, we have,

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' ρ − + ρ − + − β ( p 1 ) / 2 ( p 1 ) / 2 × Y 2 P ∫X > 0 ∫Y > 0 X 1 −1 −1 ' ' 2 − − F (α, β, β ; γ, γ ; X, P YP 2 )dXdY 2

 ( 2.13)

which, on the application of the same transformation as in eq.(2.8) above and then using the definition (1.2) yields the same result as in eq.(2.12) above. THEOREM 2.4: F3 (α , α ' , β, β' ; γ + γ ' ;− X,−Y ) ' Γp ( γ + γ ) I U γ − (p +1) / 2 I U γ ' −(p +1) / 2 F ( , ; ; U 1 2 XU 1 2 ) = − × 2 1αβ γ− ' ) ∫0 ( ) ( Γp γ Γp γ

1 1 ' ' ' 2 F [ , ; ; (I U) Y(I − U) 2 ]dU 2 1α β γ − − for 0 < U < I and for Re (γ , γ ' ) > (p − 1) / 2.

 (2.14)

PROOF: Taking the M-transform of the right side of eq. (2.14) with respect to the variables X and Y and the parameters ρ and ρ respectively, we get,

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ρ − ( p +1) / 2 Y ρ − ( p +1) / 2 F (α, β; γ;− U 12 XU 1 2 ) × X 2 ∫X >0 ∫Y >0 1 2 1 1 1 F [α ' , β' ; γ '; −(I − U ) 2 Y (I − U ) 2 ]dXdY  (2.15) 2 1 Applying the transformations,

1 1 1 1 X = U 2 XU 2 , Y = ( I − U) 2 Y (I − U ) 2 with dX = U (p +1) / 2 dX, 1 1 1 (p +1) / 2 dY1 = I − U dY and X1 = U X , Y1 = I − U Y

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to the above expression and then applying eq. (2.3.5) page 38 of Mathai [4] leads us to,

' ' ' Γp ( γ)Γp (α − ρ2 )Γp (β − ρ 2 )Γp ( γ ) U − ρ1 I − U −ρ2 × ' Γp (α)Γp (β)Γp ( γ − ρ1)Γp (α ) Γp (ρ1)Γp (ρ 2 )Γp (α − ρ1)Γp (β − ρ1) Γp (β ' )Γp ( γ ' − ρ 2 )

 ( 2.16)

Substituting this expression on the right side of eq.(2.14) and then integrating out the variable U in the resulting expression by using a type -1 Beta integral results in M ( F )

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(definition (1.3) ). THEOREM 2.5: For p =2, ' P − α F3[α, (α ' + 1) / 2, β, ( 2α ' + 1) / 4; γ;− X, −4 P −1YP −1 ) ' 1 − tr (PT) T α −( p +1) / 2 Ξ (α, β; γ;−X,−TYT ' )dT e = 2 ' ∫T >0 Γp (α )

 (2. 17)

where Re (α ' ) > ( p − 1) / 2. PROOF: Taking the M-transform of the right side of eq. (2.17) with respect to the variables X and Y and the parameters ρ and ρ respectively, we have, 1 2 ρ − ( p +1) / 2 Y ρ 2 − ( p +1) / 2 × ∫X >0 ∫Y >0 X 1 Ξ 2 (α, β; γ;− X ,− TYT ' )dXdY  ( 2.18)

p 1 2 Applying the transformation Y = TYT' with dY = T + dY and Y = T Y to 1 1 1 the above expression and then using the definition (1.8) produces,

Γp ( γ )Γp (ρ1)Γp (ρ 2 )Γp (α − ρ1)Γp (β − ρ1 ) 2 − ρ T 2 Γp (α)Γp (β)Γp ( γ − ρ1 − ρ 2 )

 ( 2.19)

Substituting this expression on the right side of eq. (2.17) and then integrating out T in the resulting expression by using a Gamma integral leads us to,

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Γp (γ )Γp (ρ1)Γp (ρ 2 )Γp (α − ρ1)Γp (β − ρ1)Γp (α ' − 2ρ 2 ) ' P − (α − 2ρ 2 )  (2.20) ' Γp (α )Γp (α)Γp (β)Γp ( γ − ρ1 − ρ 2 ) Now taking the M-transform of the left side of eq. (2.17) with respect to the variables X and Y and the parameters ρ and ρ respectively, we get,

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' ( p 1 ) / 2 ( p 1 ) / 2 ρ − + ρ − + − α Y 2 P × ∫X > 0 ∫Y > 0 X 1 F [α, (α ' + 1) / 2, β, (2α ' + 1) / 4; γ; − X,−4P −1YP −1)dXdY 3

 ( 2.21)

Making use of the transformation

Y2 = 4 P −1YP −1 with dY2 = 4 p( p +1) / 2 P − ( p +1) dY and Y2 = 4 p P − 2 Y in the

above expression and then using the definition (1.3) along with the observation that for p=2,

' ' ' − pρ 2 Γp [(α + 1) / 2 − ρ 2 ]Γp [(2α + 1) / 4 − ρ 2 ] Γp (α − 2ρ 2 ) 4  (2.22) = ' ' ' Γp (α ) Γp [(α + 1) / 2]Γp [(2α + 1) / 4] from eq. (6.13) page 84 of Mathai [4], finally leads us to the same result as in eq. (2.20) above. It is to be noted that this result is different from the corresponding result in the scalar case. THEOREM 2.6:

− 12 − 1 2 − 1 2 − 12 − α ' P F4 (α, β; γ, γ ; − P XP ,− P YP ) 1 1 1 1 1 ( p 1 ) / 2 α − + tr ( PT ) ' − 2 2 2 e T = Ψ2 (β; γ , γ ;−T XT ,−T YT 2 )dT  ( 2.23) ∫ T 0 > Γp (α) for Re (α) > (p − 1) / 2. PROOF: Taking the M-transform of the right side of eq. (2.23) with respect to the variables X and Y and the parameters ρ and ρ respectively, we have,

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ρ − ( p +1) / 2 Y ρ 2 − ( p +1) / 2 × ∫X > 0 ∫Y > 0 X 1 1 1 1 1 Ψ2 (β; γ , γ ';− T 2 XT 2 ,−T 2 YT 2 )dXdY

 (2.24)

which, under the transformations

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1 1 1 ( p 1) / 2 (p 1) / 2 2 2 2 X1 = T XT , Y1 = T YT 2 with dX1 = T + dX, dY1 = T + dY and X1 = T X , Y1 = T Y and then using the definition (1.7) yields,

Γp ( γ )Γp ( γ ' ) Γp (ρ1)Γp (ρ 2 )Γp (β − ρ1 − ρ 2 ) T − ρ1 − ρ 2 Γp (β)Γp ( γ − ρ1)Γp ( γ ' − ρ 2 )

 (2.25)

Substituting this expression on the right side of eq.(2.23) and then integrating out T in the resulting expression by using a Gamma integral produces,

Γp ( γ ' )Γp ( γ )Γp (ρ1 )Γp (ρ 2 ) Γp (α − ρ1 − ρ 2 )Γp (β − ρ1 − ρ 2 ) P − (α − ρ1 − ρ 2 )  (2.26) ' Γp (α)Γp (β)Γp ( γ − ρ1)Γp ( γ − ρ 2 ) Now, taking the M-transform of the left side of eq.(2.23) with respect to the variables X and Y and the parameters ρ and ρ respectively, we get,

∫X > 0 ∫Y > 0

1 2 X ρ1 − ( p +1) / 2 Y ρ 2 − (p +1) / 2 ×

− 1 2 − 12 − 12 − 1 2 − α ' − − α β γ γ P F4 ( , ; , ; P XP , P YP )dXdY

 ( 2.27)

which, under the transformations

− 12

− 12

− 12

− 12

with dX = P − ( p +1) / 2 dX, 2 ( p 1) / 2 1 1 dY2 = P − + dY and X 2 = P − X , Y2 = P − Y X =P 2

XP

,Y = P 2

YP

and then using the definition (1.4) leads us to the same result as in eq. (2.26). References 1. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F. G. (1954). Tables of Integral Transforms, Vol. I, McGraw Hill, New York, Toronto and London.

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2. Exton H. (1976). Multiple Hypergeometric Functions and Applications, Ellis Horwood Limited, Publishers, Chichester. 3. Mathai A.M. (1992). Jacobians of Matrix Transformations I, Centre for Mathematical Sciences, Trivandrum, India. 4. Mathai A.M. (1993). Hypergeometric Functions of Several Matrix Arguments, Centre for Mathematical Sciences, Trivandrum, India. 5. Mathai A.M. (1993). Appell's and Humbert's Functions of Matrix Arguments, Linear Algebra and its Applications; 183, pp. 201-221. 6. Mathai A.M. (1995). Special Functions of Matrix Arguments-III; Proceedings of the National Academy of Sciences, India; LXV (IV) pp. 367-393. 7. Saxena R.K., Sethi P.L. & Gupta O.P. (1997). Appell's Functions of Matrix Arguments. Indian J. Pure Appl. Math.; 28, no. 3, pp. 371-380. 8. Srivastava H.M., Karlsson P.W. (1985). Multiple Gaussian Hypergeometric Series. Ellis Horwood Limited, Publishers, Chichester. 9. Upadhyaya Lalit Mohan, Dhami H.S.(Nov.2001). Matrix Generalizations of Multiple Hypergeometric Functions, # 1818 IMA Preprints Series, University of Minnesota, Minneapolis, U.S.A.