IOSR Journal of Applied Chemistry (IOSR-JAC) e-ISSN: 2278-5736.Volume 10, Issue 2 Ver. I (Feb. 2017), PP 27-29 www.iosrjournals.org
Partial Derivatives Involving Generalized I–Function of Two Variables Shanti Swaroop Dubey1, Dr. S. S. Shrivastava2 1
ITM University, Raipur (C. G.) Institute for Excellence in Higher Education, Bhopal (M. P.)
2
Abstract: The aim of this paper is to derive partial derivatives involving generalized I–function of two variables. I. Introduction The generalized I–function of two variables introduced by Goyal and Agrawal [1], will be defined and represented as follows: [(a j :ιj ,A j )1,n ],[(a ji :ιji ,A ji )n +1,p i ] m,n:m ,n 1 :m 2 ,n 2 [x | ,q ′ :r ′ :p ′′ ,q ′′ :r ′′ y [(b j :βj ,B j )1,n ],[(b ji :βji ,B ji )1,q i ] ′ i i i i i i :[(c j ;γj )1,n 1 ],[(c ′ ;γ ′ )n 1 +1,p ′ ];[(e j ;E j )1,n 2 ],[(e ′′ ;E ′′ )n 2 +1,p ′′ ] ji ji ji ji
I[xy ] = Ip ,q :r:p1
i
i
:[(d j ;δj )1,m 1 ],[(d ′ ;δγ ′ )m 1 +1,q ′ ];[(f j ;F j )1,m 2 ],[(f ′′ ;F ′′ )m 2 +1,q ′′ ] ji ji ji ji i i
=
1 (2πω)2 L 1 L 2
]
ϕ1 Ξ, Ρ θ2 Ξ θ3 (Ρ) x Ξ y Ρ dΞdΡ,
(1)
where ϕ1 Ξ, Ρ =
r [ i=1
m n j=1 Γ(b j âˆ’Î˛j Ξ−B j Ρ) j=1 Γ(1−a j +Îąj Ξ+A j Ρ) pi qi Γ(a ji âˆ’Îąji Ξ−A ji Ρ) j=1 Γ(1−b ji +βji Ξ+B ji Ρ) j=n +1
,
θ2 Ξ =
m1 n1 j=1 Γ(d j −δj Ξ) j=1 Γ(1−c j +Îłj Ξ) q p ′ r′ [ i′ i Γ(1−d ji ′ +δji ′ Ξ) j=n Γ(c ji ′ âˆ’Îłji ′ Ξ) i ′ =1 j=m 1 +1 1 +1
θ3 Ρ =
m2 n2 j=1 Γ(f j −F j Ρ) j=1 Γ(1−e j +E j Ρ) q p ′′ r ′′ [ i ′′ i Γ(1−f ji ′′ +F ji ′′ Ρ) j=n Γ(e ji ′′ −E ji ′′ Ρ) i ′′ =1 j=m 2 +1 2 +1
,
,
x and y are not equal to zero, and an empty product is interpreted as unity p i, pi´, pi´´, qi, qi´, qi´´, m, n, n1, n2, nj and mk are non negative integers such that pi ď‚łď€ n ď‚łď€ 0, pi´ ď‚łď€ n1 ď‚łď€ 0, pi´´ ď‚łď€ n2 ď‚łď€ 0, qi ď‚łď€ m >ď€ 0, qi´ ď‚ł 0, qi´´ ď‚ł 0, (i = 1, ‌, r; i´ = 1, ‌, r´; i´´ = 1, ‌, r´´; k = 1, 2) also all the A’s, ď Ąâ€™s, B’s, ď ˘â€™s, ď §â€™s, ď ¤â€™s, E’s and F’s are assumed to be positive quantities for standardization purpose; the definition of I-function of two variables given above will however, have a meaning even if some of these quantities are zero. The contour L 1 is in the ď ¸ď€plane and runs from – ď ˇď‚Ľ to + ď ˇď‚Ľ, with loops, if necessary, to ensure that the poles of ď ‡ď€¨djď€ď ¤jď ¸) (j = 1, ..........., m1) lie to the right, and the poles of ď ‡ď€ ď€¨ď€ąď€ ď€ď€ cj ď€Ťď€ ď §jď ¸) (j = 1, ..., n1), ď ‡ď€ ď€¨ď€ ď€ąď€ ď€ď€ aj + ď Ąjď ¸ď€ + Ajď ¨) (j = 1, ..., n) to the left of the contour. The contour L2 is in the ď ¨ď€plane and runs from – ď ˇď‚Ľ to + ď ˇď‚Ľ, with loops, if necessary, to ensure that the poles of ď ‡ď€ ď€¨fj ď€ Fjď ¨) (j=1,....., n2) lie to the right, and the poles of ď ‡ď€ ď€¨ď€ąď€ ď€ď€ ej  Ejď ¨) (j = 1, ..., m2), ď ‡ď€ ď€¨ď€ ď€ąď€ ď€ď€ aj + ď Ąjď ¸ď€ + Ajď ¨) (j = 1, ..., n) to the left of the contour. Also
đ?‘ˆ= đ?‘‰=−
�′ =
pi j=1 Îąji
+
p′ i γ j=1 ji ′
�′ =
pi j=1 A ji
+
p ′′ i E j=1 ji ′′
pi j=n+1 Îąji
−
pi j=n+1 A ji
qi j=m+1 βji
−
−
m1 j=1 δj
+
qi j=m+1 Bji
qi j=1 βji
−
−
qi j=1 Bji
−
m2 j=1 Fj
q′ i δ j=1 ji ′
− −
q ′′ i Fδji ′ j=1
q′ i δ j=m 1 +1 ji ′
−
< 0,
+
q â&#x20AC;˛â&#x20AC;˛ i F j=m 21 +1 ji â&#x20AC;˛â&#x20AC;˛
< 0, n1 j=1 Îłj
+
â&#x2C6;&#x2019;
n2 j=1 Ej
pâ&#x20AC;˛ i Îł j=n 1 +1 ji â&#x20AC;˛
â&#x2C6;&#x2019;
> 0,
p â&#x20AC;˛â&#x20AC;˛ i E j=n 2 +1 ji â&#x20AC;˛â&#x20AC;˛
(2) > 0,
(3)
and |arg x| < ½ Uď °, |arg y| < ½ Vď °ď&#x20AC;Žď&#x20AC;
DOI: 10.9790/5736-1002012729
www.iosrjournals.org
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