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 ,ι j ;A j )1,p 1 : c j ,γ 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
ϕ1 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 âˆ’Î˛ 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€
ϕ1 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&#x20AC;Žď&#x20AC; ď&#x20AC; ď&#x20AC; ď&#x20AC; ď&#x20AC; ď&#x20AC; DOI: 10.9790/5728-1301054547
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