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
www.iosrjournals.org
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On Double Integrals Involving Generalized Hâ&#x20AC;&#x201C;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] (â&#x2C6;&#x2019;n)m u u=0 u!
Snm x =
X u An,u , (n = 0, 1, 2, â&#x20AC;Ś),
(4)
where m is arbitrary positive integer, and the coefficients A n,u (n, u â&#x2030;Ľ 0) are arbitrary constants, real or complex.
III.
Main Result
In this paper we will establish the following integral: 1
1
x Îťâ&#x2C6;&#x2019;1 (1 â&#x2C6;&#x2019; x)Îźâ&#x2C6;&#x2019;1 y Ď â&#x2C6;&#x2019;1 (1 â&#x2C6;&#x2019; y)aâ&#x2C6;&#x2019;2Ď (1 + ty)Ď â&#x2C6;&#x2019;aâ&#x2C6;&#x2019;1 0
0
a,b; . 2F 1[1+aâ&#x2C6;&#x2019;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&#x201A; (1â&#x2C6;&#x2019;x)ν {
[z 2
1 â&#x2C6;&#x2019; 2x; k Snm [cy Îł 1 + ty
|(b ,β ;B ) j
j
a 2
Ď&#x20AC; Î&#x201C; 1+a Î&#x201C; 1+ â&#x2C6;&#x2019; b s! (Îą+β+s+1)kj (â&#x2C6;&#x2019;s)j c u (Îą+1)kj j! 4 1+t
Îłu
1â&#x2C6;&#x2019;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 (1â&#x2C6;&#x2019;y )2
a 2 a 4 1+t â&#x2C6;&#x2019;Ď Î&#x201C; 1+2 Î&#x201C;(1+aâ&#x2C6;&#x2019;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
â&#x2C6;&#x2019;2Îł
]
]dx dy
s (â&#x2C6;&#x2019;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 â&#x2C6;&#x2019;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 â&#x2C6;&#x2019; Îź, ν , 1 â&#x2C6;&#x2019; Îť â&#x2C6;&#x2019; kj, Ď&#x201A; , 1 â&#x2C6;&#x2019; Ď â&#x2C6;&#x2019; Îłu, T , cj , Îłj B=
1 2
a
a
2
2
1,p 2
+ â&#x2C6;&#x2019; Ď â&#x2C6;&#x2019; Îłu, T , 1 + â&#x2C6;&#x2019; b â&#x2C6;&#x2019; Ď â&#x2C6;&#x2019; Îłu, T , dj , δj
, (1 + a â&#x2C6;&#x2019; b â&#x2C6;&#x2019; Ď â&#x2C6;&#x2019; Îłu, T)
1,q 2
, 1 â&#x2C6;&#x2019; Îť â&#x2C6;&#x2019; Îź â&#x2C6;&#x2019; kj, Ď&#x201A; + 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,uâ&#x2030;Ľ0) are arbitrary constants, real or complex. (ii) | arg z1 | < ½ Uď °, | arg z2 | < ½ Vď °ď&#x20AC;Źď&#x20AC; where U and V are given in (2) and (3) respectively. (iii) Re 1 + a â&#x2C6;&#x2019; b > 0, đ?&#x2018;&#x2026;đ?&#x2018;&#x2019; Îą > â&#x2C6;&#x2019;1, đ?&#x2018;&#x2026;đ?&#x2018;&#x2019; β > â&#x2C6;&#x2019;1, đ?&#x2018;Ą > â&#x2C6;&#x2019;1, Îł â&#x2030;Ľ 0, Re Îť + Ď&#x201A;Ξ > 0, đ?&#x2018;&#x2026;đ?&#x2018;&#x2019; Ď + νΞ > 0, Re Ď + Îłu + TΞ > 0, Re 1 + a â&#x2C6;&#x2019; 2Ď â&#x2C6;&#x2019; 2Îłu â&#x2C6;&#x2019; 2TΞ > 0. (iv) Re(1â&#x2C6;&#x2019;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&#x201A;Ξ â&#x2C6;&#x2019;1 x 0
.
1 Ď +Îłu+TΞâ&#x2C6;&#x2019;1 y 0
a,b; . 2F 1[1+a+b;
[n/m] (â&#x2C6;&#x2019;n)m u u=0 u!
1â&#x2C6;&#x2019;x
c u An,u
Îź+νΞ â&#x2C6;&#x2019;1 Îą,β Js
(â&#x2C6;&#x2019;1) 4Ď&#x20AC; 2
L1 L2
Ď&#x2022;1 Ξ, Ρ θ2 (Ξ)θ3 (Ρ)
1 â&#x2C6;&#x2019; 2x; k dx
(1 â&#x2C6;&#x2019; y)aâ&#x2C6;&#x2019;2TΞâ&#x2C6;&#x2019;2Îłuâ&#x2C6;&#x2019;2Ď (1 + ty)Ď â&#x2C6;&#x2019;a+Îłu+Tâ&#x2C6;&#x2019;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 Hâ&#x20AC;&#x201C;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 Îťâ&#x2C6;&#x2019;1 (1 â&#x2C6;&#x2019; x)Îźâ&#x2C6;&#x2019;1 y Ď â&#x2C6;&#x2019;1 (1 â&#x2C6;&#x2019; y)aâ&#x2C6;&#x2019;2Ď (1 + ty)Ď â&#x2C6;&#x2019;aâ&#x2C6;&#x2019;1 0
0
a,b; . 2F 1[1+aâ&#x2C6;&#x2019;b;
(1+t)y 1+ty
] Js
M,N . HP,Q [zx Ď&#x201A; (1 â&#x2C6;&#x2019; x)ν {
=
.
ι,β
1 â&#x2C6;&#x2019; 2x; k Snm [cy Îł 1 + ty
y(1+ty ) T (a j ,Îą j )1,P } |(b ,β ) ]dx (1â&#x2C6;&#x2019;y)2 j j 1,Q
a 2 a 4 1+t â&#x2C6;&#x2019;Ď Î&#x201C; 1+ Î&#x201C;(1+aâ&#x2C6;&#x2019;b)(a+1)ks 2
a 2
Ď&#x20AC; Î&#x201C; 1+a Î&#x201C; 1+ â&#x2C6;&#x2019; b s!
(Îą+β+s+1)kj (â&#x2C6;&#x2019;s)j c u (Îą+1)kj j! 4 1+t Îł u
[n/m] u=0
Îł
1â&#x2C6;&#x2019;y
â&#x2C6;&#x2019;2Îł
]
dy s (â&#x2C6;&#x2019;n)m u j=0 u!
M+2,N+3 An,u HP+4,Q+3 [z 4 1 + t
â&#x2C6;&#x2019;T A | B ],
(6)
where A = 1 â&#x2C6;&#x2019; Îź, ν , 1 â&#x2C6;&#x2019; Îť â&#x2C6;&#x2019; kj, Ď&#x201A; , 1 â&#x2C6;&#x2019; Ď â&#x2C6;&#x2019; Îłu, T , a j , Îąj B=
1 2
a
a
2
2
1,P
+ â&#x2C6;&#x2019; Ď â&#x2C6;&#x2019; Îłu, T , 1 + â&#x2C6;&#x2019; b â&#x2C6;&#x2019; Ď â&#x2C6;&#x2019; Îłu, T , bj , βj
, (1 + a â&#x2C6;&#x2019; b â&#x2C6;&#x2019; Ď â&#x2C6;&#x2019; Îłu, T)
1,Q
, 1 â&#x2C6;&#x2019; Îť â&#x2C6;&#x2019; Îź â&#x2C6;&#x2019; kj, Ď&#x201A; + 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,uâ&#x2030;Ľ0) are arbitrary constants, real or complex. Q (ii) Ί = Nj=1 Îąj â&#x2C6;&#x2019; Pj=n+1 Îąj + M j=1 βj â&#x2C6;&#x2019; j=n+1 βj > 0, and |arg z| < (1/2)ΊĎ&#x20AC;. (iii) Re 1 + a â&#x2C6;&#x2019; b > 0, đ?&#x2018;&#x2026;đ?&#x2018;&#x2019; Îą > â&#x2C6;&#x2019;1, đ?&#x2018;&#x2026;đ?&#x2018;&#x2019; β > â&#x2C6;&#x2019;1, đ?&#x2018;Ą > â&#x2C6;&#x2019;1, Îł â&#x2030;Ľ 0, Re Îť + Ď&#x201A;Ξ > 0, đ?&#x2018;&#x2026;đ?&#x2018;&#x2019; Ď + νΞ > 0, Re Ď + Îłu + TΞ > 0, Re 1 + a â&#x2C6;&#x2019; 2Ď â&#x2C6;&#x2019; 2Îłu â&#x2C6;&#x2019; 2TΞ > 0. (iv) Re(1â&#x2C6;&#x2019;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