On Double Integrals Involving Generalized H–Function of Two Variables

Page 1

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue I Ver. V (Jan. - Feb. 2017), PP 45-47 www.iosrjournals.org

On Double Integrals Involving Generalized H–Function of Two Variables Mehphooj Beg1 & Dr. S. S. Shrivastava2 1

VITS, Satna (M. P.) Institute for Excellence in Higher Education Bhopal (M. P.)

2

Abstract: The aim of this paper is to derive a double integrals involving generalized H–function of two variables.

I. Introduction The generalized H–function of two variables is given by Shrivastava, H. S. P. [4] and defined as follows: (a j ,Îą j ;A j )1,p 1 : c j ,Îł j : e j ,E j m ,n ;m ,n ;m ,n 3 x 1,p 1,p 3 [y |(b ,β ;B ) : d ,δ 2 : f ,F ] 1,q j j j j j 1,q j j 1,q 1

2 3 Hp 11,q 11;p 2 ,q2 2 ;p 3 ,q 3

2

=

−1 4Ď€ 2 L 1 L 2

3

Ď•1 Ξ, Ρ θ2 (Ξ)θ3 (Ρ)x Ξ y Ρ dΞdΡ

(1)

where n1 m1 j=1 Γ(1−a j +Îą j Ξ+A j Ρ) j=1 Γ(b j −β j Ξ−B j Ρ) p1 q1 Γ(a j âˆ’Îą j Ξ−A j Ρ) j=1 Γ(1−b j +β j Ξ+B j Ρ) j=n 1 +1

ď€

Ď•1 Ξ, Ρ =

ď€

θ2 Ξ =

m2 n2 j=1 Γ(d j −δ j Ξ) j=1 Γ(1−c j +Îł j Ξ) q2 p2 Γ(1−d j +δ j Ξ) j=n Γ(c j âˆ’Îł j Ξ) j=m 2 +1 2 +1

ď€

θ2 Ξ =

m3 n3 j=1 Γ(f j −F j Ρ) j=1 Γ(1−e j +E j Ρ) q3 p3 Γ(1−f j +F j Ρ) j=n Γ(e j −E j Ρ) j=m 3 +1 3 +1

,ď€

ď€ ď€

x and y are not equal to zero, and an empty product is interpreted as unity p i, qi, ni and mj are non negative integers such that pi ≼ ni ≼ 0, qi ≼ 0, qj ≼ mj ≼ 0, (i = 1, 2, 3; j = 2, 3). Also, all the A’s, ι’s, B’s, β’s, γ’s, ď ¤â€™s, E’s, and F’s are assumed to the positive quantities for standardization purpose. The contour L1 is in the ď ¸-plane and runs from – i∞ to + i∞, with loops, if necessary, to ensure that the poles of ď ‡(dj - ď ¤jď ¸) (j = 1, ..., m2) lie to the right, and the poles of ď ‡(1 – cj + Îłjď ¸) (j = 1, ..., n2), ď ‡(1 – aj + Îąjď ¸+ Ajď ¨) (j = 1, ..., n1) to the left of the contour. The contour L2 is in the ď ¨-plane and runs from – i∞ to + i∞, with loops, if necessary, to ensure that the poles of ď ‡(fj – Fjď ¨) (j = 1, ..., m3) lie to the right, and the poles of ď ‡(1 – ej + Ejď ¨) (j = 1, ..., n3), ď ‡(1 – aj + Îąjď ¸ + Ajď ¨) (j = 1, ..., n1) to the left of the contour. The generalized H-function of two variables given by (1) is convergent if đ?‘ˆ= −

�1 � =1 ��

đ?‘?1 đ?‘— =đ?‘› 1 +1 đ?›źđ?‘—

�=

đ?‘›1 đ?‘— =1 đ??´đ?‘—

+ − +

�1 � =1 ��

+

đ?‘ž1 đ?‘— =đ?‘š 1 +1 đ?›˝đ?‘— đ?‘š1 đ?‘— =1 đ??ľđ?‘—

+

�2 � =1 ��

−

�2 � =1 ��

đ?‘?2 đ?‘— =đ?‘› 2 +1 đ?›žđ?‘—

đ?‘›3 đ?‘— =1 đ??¸đ?‘—

−

+

+

−

�2 � =� 2 +1 �� ;

(2)

đ?‘š3 đ?‘— =1 đ??šđ?‘—

đ?‘?1 đ?‘— =đ?‘› 1 +1 đ??´đ?‘—

−

đ?‘ž1 đ?‘— =đ?‘š 1 +1 đ??ľđ?‘—

−

đ?‘?3 đ?‘— =đ?‘› 3 +1 đ??¸đ?‘—

−

đ?‘ž3 đ?‘— =đ?‘š 3 +1 đ??šđ?‘— ,

(3) where | arg x | < ½ Uď °, | arg y | < ½ Vď °ď€Žď€ ď€ ď€ ď€ ď€ ď€ DOI: 10.9790/5728-1301054547

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On Double Integrals Involving Generalized H–Function Of Two Variables II.

Result Required

The following result are required in our present investigation: From Shrivastava [3, p.1, Eq. (1)]: [n/m] (−n)m u u=0 u!

Snm x =

X u An,u , (n = 0, 1, 2, ‌),

(4)

where m is arbitrary positive integer, and the coefficients A n,u (n, u ≼ 0) are arbitrary constants, real or complex.

III.

Main Result

In this paper we will establish the following integral: 1

1

x Îťâˆ’1 (1 − x)Îźâˆ’1 y Ď âˆ’1 (1 − y)a−2Ď (1 + ty)Ď âˆ’a−1 0

0

a,b; . 2F 1[1+a−b;

m ,n ;m ,n ;m ,n 3

. Hp 11,q 11;p 2 ,q2 2 ;p2 3 ,q33

=

(1+t)y 1+ty

] Js

ι,β

z 1 x Ď‚ (1−x)ν {

[z 2

1 − 2x; k Snm [cy Îł 1 + ty

|(b ,β ;B ) j

j

a 2

Ď€ Γ 1+a Γ 1+ − b s! (Îą+β+s+1)kj (−s)j c u (Îą+1)kj j! 4 1+t

Îłu

1−y

y (1+ty ) T : e j ,E j } (a j ,Îą j ;A j )1,p 1 : c j ,Îł j 1,p 2 1,p 3 (1−y )2

a 2 a 4 1+t âˆ’Ď Î“ 1+2 Γ(1+a−b)(a+1)ks

.

Îł

j 1,q 1 : d j ,δ j 1,q : f j ,F j 1,q 2 3

[n/m] u=0

−2Îł

]

]dx dy

s (−n)m u j=0 u!

(a j ,Îą j ;A j )1,p 1 :A: e j ,E j m ,n ;m +2,n +3;m ,n 3 z 1 4 1+t −T 1,p 3 [z 2 |(b ,β ;B ) :B: f ,F ], j j j 1,q 1 j j 1,q 3

2 2 3 An,u Hp 11,q 11;p 2 +4,q 2 +3;p 3 ,q 3

(5)

where A = 1 − Îź, ν , 1 − Îť − kj, Ď‚ , 1 − Ď âˆ’ Îłu, T , cj , Îłj B=

1 2

a

a

2

2

1,p 2

+ − Ď âˆ’ Îłu, T , 1 + − b − Ď âˆ’ Îłu, T , dj , δj

, (1 + a − b − Ď âˆ’ Îłu, T)

1,q 2

, 1 − Îť − Îź − kj, Ď‚ + ν .

The integral (5) is valid if the following sets of (sufficient) conditions are satisfied: (i) m is arbitrary positive integer, and the coefficient An,u (n,u≼0) are arbitrary constants, real or complex. (ii) | arg z1 | < ½ Uď °, | arg z2 | < ½ Vď °ď€Źď€ where U and V are given in (2) and (3) respectively. (iii) Re 1 + a − b > 0, đ?‘…đ?‘’ Îą > −1, đ?‘…đ?‘’ β > −1, đ?‘Ą > −1, Îł ≼ 0, Re Îť + Ď‚Ξ > 0, đ?‘…đ?‘’ Ď + νΞ > 0, Re Ď + Îłu + TΞ > 0, Re 1 + a − 2Ď âˆ’ 2Îłu − 2TΞ > 0. (iv) Re(1−2b) > 0. Proof: To establish (5), we use the series representation of Snm [x] as given by (4) and for the generalized Hfunction of two variables we use Mellin-Barnes types of contour integral as given in (1), on the left-hand side of (5), change the order of integration and summation (which is justified under the conditions given with (5)), we then obtain Left-hand side of (5) = .

1 Îť+Ď‚Ξ âˆ’1 x 0

.

1 Ď +Îłu+TΞ−1 y 0

a,b; . 2F 1[1+a+b;

[n/m] (−n)m u u=0 u!

1−x

c u An,u

Îź+νΞ âˆ’1 Îą,β Js

(−1) 4Ď€ 2

L1 L2

Ď•1 Ξ, Ρ θ2 (Ξ)θ3 (Ρ)

1 − 2x; k dx

(1 − y)a−2TΞ−2Îłu−2Ď (1 + ty)Ď âˆ’a+Îłu+T−1

(1+t)y 1+ty

]dy]x Ξ y Ρ dΞdΡ

DOI: 10.9790/5728-1301054547

www.iosrjournals.org

46 | Page


On Double Integrals Involving Generalized H–Function Of Two Variables Now using known results [1, p.118, Eq.(3.3.1)] and [2, p.254, Eq.(2.1)], and interpreting the resulting contour integral as the generalized H-function of two variables, we once get the right-hand side of (5).

IV. Special Cases On specializing the parameters in (5), we get following integral in terms of H-function: 1

1

x Îťâˆ’1 (1 − x)Îźâˆ’1 y Ď âˆ’1 (1 − y)a−2Ď (1 + ty)Ď âˆ’a−1 0

0

a,b; . 2F 1[1+a−b;

(1+t)y 1+ty

] Js

M,N . HP,Q [zx Ď‚ (1 − x)ν {

=

.

ι,β

1 − 2x; k Snm [cy Îł 1 + ty

y(1+ty ) T (a j ,Îą j )1,P } |(b ,β ) ]dx (1−y)2 j j 1,Q

a 2 a 4 1+t âˆ’Ď Î“ 1+ Γ(1+a−b)(a+1)ks 2

a 2

Ď€ Γ 1+a Γ 1+ − b s!

(Îą+β+s+1)kj (−s)j c u (Îą+1)kj j! 4 1+t Îł u

[n/m] u=0

Îł

1−y

−2Îł

]

dy s (−n)m u j=0 u!

M+2,N+3 An,u HP+4,Q+3 [z 4 1 + t

−T A | B ],

(6)

where A = 1 − Îź, ν , 1 − Îť − kj, Ď‚ , 1 − Ď âˆ’ Îłu, T , a j , Îąj B=

1 2

a

a

2

2

1,P

+ − Ď âˆ’ Îłu, T , 1 + − b − Ď âˆ’ Îłu, T , bj , βj

, (1 + a − b − Ď âˆ’ Îłu, T)

1,Q

, 1 − Îť − Îź − kj, Ď‚ + ν .

The integral (6) is valid if the following sets of (sufficient) conditions are satisfied: (i) m is arbitrary positive integer, and the coefficient An,u (n,u≼0) are arbitrary constants, real or complex. Q (ii) Ί = Nj=1 Îąj − Pj=n+1 Îąj + M j=1 βj − j=n+1 βj > 0, and |arg z| < (1/2)ΊĎ€. (iii) Re 1 + a − b > 0, đ?‘…đ?‘’ Îą > −1, đ?‘…đ?‘’ β > −1, đ?‘Ą > −1, Îł ≼ 0, Re Îť + Ď‚Ξ > 0, đ?‘…đ?‘’ Ď + νΞ > 0, Re Ď + Îłu + TΞ > 0, Re 1 + a − 2Ď âˆ’ 2Îłu − 2TΞ > 0. (iv) Re(1−2b) > 0.

References [1]. [2]. [3]. [4].

Gupta, Rajni: A Study of Generalized Fractional Operator and Multivariable Special Functions with Applications, Ph. D. thesis, Univ. of Rajsthan, India, 1988. Rathie, N.: Integrals involving H-function, Vijnana Parishad Anusandhan Patrika, 22 (1979), 253 - 258. Shrivastava, H. M.: A contour integral involving Fox's H-function, Indian J. Math. 14 (1972), 1 - 6. Srivastava, H. S. P.: H-function of two variables I, Indore Univ., Res. J Sci. 5(1-2), p.87-93, (1978).

DOI: 10.9790/5728-1301054547

www.iosrjournals.org

47 | Page


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