IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. I (Jan. - Feb. 2017), PP 23-25 www.iosrjournals.org
On Some New Linear Generating Relations Involving I-Function of Two Variables Raghunayak Mishra ,Dr. S. S. Srivastava 1
Department of Mathematics Narayan Degree College Patti, Pratapgarh (U.P.) 2 Institute for Excellence in Higher Education Bhopal (M.P.)
Abstract: The aim of this research paper is to establish some linear generating relations involving I-function of two variables.
I. Introduction The I–function of two variables introduced by Sharma & Mishra [2], will be defined and represented as follows: [(a j :ι j ,A j )1,n ],[(a ji :ι ji ,A ji )n +1,p i ]
0,n:m ,n :m ,n
1 1 2 2 x I[xy ] = Ip ,q :r:p ′ ′′ [y |[(b ′ ,q ′ :r :p ′′ ,q ′′ :r i
i
i
i
ji :β ji ,B ji )1,q i ] i i :[(c j ;γ j )1,n 1 ],[(c ji ′ ;γ ji ′ )n 1 +1,p ′ ];[(e j ;E j )1,n 2 ],[(e ji ′′ ;E ji ′′ )n 2 +1,p ′′ ] i i ] :[(d j ;δ j )1,m 1 ],[(d ji ′ ;δγ ji ′ )m 1 +1,q ′ ];[(f j ;F j )1,m 2 ],[(f ji ′′ ;F ji ′′ )m 2 +1,q ′′ ] i
=
1 (2πω )2 L 1 L 2
i
ϕ1 Ξ, Ρ θ2 Ξ θ3 (Ρ) x Ξ y Ρ dΞdΡ,
(1)
where ϕ1 Ξ, Ρ =
r [ i=1
n j=1 Γ(1−a j +Îą j Ξ+A j Ρ) pi qi Γ(a âˆ’Îą ji Ξ−A ji Ρ) j=1 Γ(1−b ji +β ji Ξ+B ji Ρ) ji j=n +1
θ2 Ξ =
m1 n1 j=1 Γ(d j −δ j Ξ) j=1 Γ(1−c j +Îł j Ξ) q p ′ r′ [ i′ i Γ(1−d ji ′ +δ ji ′ Ξ) j=n Γ(c ji ′ âˆ’Îł ji ′ 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 pi, pi´, pi´´, qi, qi´, qi´´, 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€ 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 L1 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 §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=
�′ =
pi j=1 Îąji
+
�′ =
pi j=1 A ji
+
pi j=n+1 Îąji
−
qi j=1 βji
pi′
γ ′ j=1 ji p i ′′ j=1
+
qi j=1 βji
−
Eji ′′ − m1 j=1 δj
−
qi′
−
qi j=1 Bji
j=1
−
qi′ j=m 1 +1
δji â&#x20AC;˛ < 0,
q i â&#x20AC;˛â&#x20AC;˛ j=1
δji â&#x20AC;˛ +
Fδji â&#x20AC;˛ < 0, n1 j=1 Îłj
â&#x2C6;&#x2019;
piâ&#x20AC;˛
Îł â&#x20AC;˛ j=n 1 +1 ji
> 0, (2)
đ?&#x2018;&#x2030;=â&#x2C6;&#x2019;
pi j=n+1 A ji
â&#x2C6;&#x2019;
qi j=1 Bji
â&#x2C6;&#x2019;
m2 j=1 Fj
â&#x2C6;&#x2019;
q i â&#x20AC;˛â&#x20AC;˛ j=m 21 +1
Fji â&#x20AC;˛â&#x20AC;˛ +
n2 j=1 Ej
â&#x2C6;&#x2019;
p i â&#x20AC;˛â&#x20AC;˛ j=n 2 +1
Eji â&#x20AC;˛â&#x20AC;˛ > 0, (3)
and |arg x| < ½ Uď °, |arg y| < ½ Vď °ď&#x20AC;Žď&#x20AC; ď&#x20AC; DOI: 10.9790/5728-1301012325
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