Partial Derivatives Involving Generalized I–Function of Two Variables

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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 :Îąj ,A j )1,n ],[(a ji :Îąji ,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 ;Îłj )1,n 1 ],[(c ′ ;Îł ′ )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

]

Ď•1 Ξ, Ρ θ2 Ξ θ3 (Ρ) x Ξ y Ρ dΞdΡ,

(1)

where Ď•1 Ξ, Ρ =

r [ i=1

m n j=1 Γ(b j −βj 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  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 ′′ i F j=m 21 +1 ji ′′

< 0, n1 j=1 Îłj

+

−

n2 j=1 Ej

p′ i γ j=n 1 +1 ji ′

−

> 0,

p ′′ i E j=n 2 +1 ji ′′

(2) > 0,

(3)

and |arg x| < ½ Uď °, |arg y| < ½ Vď °ď€Žď€

DOI: 10.9790/5736-1002012729

www.iosrjournals.org

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Partial Derivatives Involving Generalized I–Function Of Two Variables II. Result Required The following result are required in our present investigation: From Rainville [2]: �Γ � = Γ(� + 1).

(4)

III. Main Result In this paper we will establish the following partial derivatives: đ?œ•

m,n:m ,n 1 :m 2 ,n 2 [x ] ,q :r ′ :p ′′ ,q ′′ :r ′′ y i i i′ i′ i i

Ip ,q :r:p1

đ?œ•đ?‘Ľ

[(a j :Îąj ,A j )1,n ],[(a ji :Îąji ,A ji )n +1,p ] m,n:m ,n 1 :m 2 ,n 2 x i ′ :p ,q :r ′′ [y |[(b :β ,B ) ,q :r j j j 1,n ],[(b ji :βji ,B ji )1,q i ] ′ ′ ′′ ′′ i i

= x −1 Ip ,q :r:p1

i

i

i

i

: 0,1 ,[(c j ;Îłj )1,n 1 ],[(c ′ ;Îł ′ )n 1 +1,p ′ ];[(e j ;E j )1,n 2 ],[(e ′′ ;E ′′ )n 2 +1,p ′′ ] ji ji ji ji i i ], :[(d j ;δj )1,m 1 ],[(d ′ ;δγ ′ )m 1 +1,q ′ ],(1,1);[(f j ;F j )1,m 2 ],[(f ′′ ;F ′′ )m 2 +1,q ′′ ] ji

ji

ji

i

ji

(5)

i

where | arg x | < ½ Uď °, | arg y | < ½ Vď °ď€Źď€ where U and V are given in (2) and (3) respectively. đ?œ•

m,n:m ,n 1 :m 2 ,n 2 [x ] ,q :r ′ :p ′′ ,q ′′ :r ′′ y i i i′ i′ i i

Ip ,q :r:p1

đ?œ•đ?‘Ś

m,n:m 1 ,n 1 :m 2 ,n 2 x [(a j :Îąj ,A j )1,n ],[(a ji :Îąji ,A ji )n +1,p i ] ′ :p ,q :r ′′ [y |[(b :β ,B ) :r:p ,q :r j j j 1,n ],[(b ji :βji ,B ji )1,q i ] i i i′ i′ i ′′ i ′′ :[(c j ;Îłj )1,n 1 ],[(c ′ ;Îł ′ )n 1 +1,p ′ ]; 0,1 ,[(e j ;E j )1,n 2 ],[(e ′′ ;E ′′ )n 2 +1,p ′′ ] ji ji ji ji

= y −1 Ip ,q

i

i

],

(6)

:[(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 ′′ ],(1,1) ji ji ji ji i

i

where | arg x | < ½ Uď °, | arg y | < ½ Vď °ď€Źď€ where U and V are given in (2) and (3) respectively. đ?œ•2 đ?œ•đ?‘Ľđ?œ•đ?‘Ś

m,n:m 1 ,n 1 :m 2 ,n 2 x ′ ′′ [y ] i i :r:p ′ ,q ′ :r :p ′′ ,q ′′ :r

Ip ,q

i

i

i

i

[(a j :Îąj ,A j )1,n ],[(a ji :Îąji ,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 : 0,1 ,[(c j ;Îłj )1,n 1 ],[(c ′ ;Îł ′ )n 1 +1,p ′ ]; 0,1 ,[(e j ;E j )1,n 2 ],[(e ′′ ;E ′′ )n 2 +1,p ′′ ] ji ji ji ji

= (xy)−1 Ip ,q :r:p1

i

i

],

:[(d j ;δj )1,m 1 ],[(d ′ ;δγ ′ )m 1 +1,q ′ ],(1,1);[(f j ;F j )1,m 2 ],[(f ′′ ;F ′′ )m 2 +1,q ′′ ],(1,1) ji ji ji ji i

(7)

i

where | arg x | < ½ Uď °, | arg y | < ½ Vď °ď€Źď€ where U and V are given in (2) and (3) respectively. Proof: To establish (5), we use for the generalized I-function of two variables Mellin-Barnes types of contour integral as given in (1), on the left-hand side of (5), change the order of integration and derivative (which is justified under the conditions given with (5)), we then obtain Left-hand side of (5) =

−1 4Ď€2

= =

Ď•

L1 L2 1 −1 4Ď€2 L 1 L 2 −1 4Ď€2

L1 L2

Ξ, Ρ θ2 Ξ θ3 Ρ

đ?œ• đ?œ•đ?‘Ľ

x Ξ y Ρ dΞdΡ

Ď•1 Ξ, Ρ θ2 Ξ θ3 Ρ Ξx Ξ−1 y Ρ dΞdΡ Ď•1 Ξ, Ρ θ2 Ξ θ3 Ρ

ΞΓ(Ξ) Ξ−1 x Γ(Ξ)

y Ρ dΞdΡ

Now using the result (4) and interpreting the resulting contour integral as the generalized I-function of two variables, we once get the right-hand side of (5). Proceeding on the similar way, the results (6) and (7) can be obtained.

IV. Special Cases On choosing m = 0 main results, we get following partial derivatives in terms of I-function of two variables: đ?œ• đ?œ•đ?‘Ľ

0,n:m ,n :m 2 ,n 2 x ′ ′′ [y ] ′ ′ :r :p ′′ ,q ′′ :r i i

1 1 Ip ,q :r:p ,q i

i

i

i

[(a j :Îąj ,A j )1,n ],[(a ji :Îąji ,A ji )n +1,p i ] 0,n:m ,n :m 2 ,n 2 [x | :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

1 1 = x −1 Ip ,q :r:p ,q

: 0,1 ,[(c j ;Îłj )1,n 1 ],[(c ′ ;Îł ′ )n 1 +1,p ′ ];[(e j ;E j )1,n 2 ],[(e ′′ ;E ′′ )n 2 +1,p ′′ ] ji ji ji ji i i :[(d j ;δj )1,m 1 ],[(d ′ ;δγ ′ )m 1 +1,q ′ ],(1,1);[(f j ;F j )1,m 2 ],[(f ′′ ;F ′′ )m 2 +1,q ′′ ] ji ji ji ji i

DOI: 10.9790/5736-1002012729

];

(8)

i

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Partial Derivatives Involving Generalized I–Function Of Two Variables đ?œ•

0,n:m ,n :m 2 ,n 2 x ′ ′′ [y ] ′ ′ :r :p ′′ ,q ′′ :r i i

1 1 Ip ,q :r:p ,q

đ?œ•đ?‘Ś

i

=

i

i

i

[(a j :Îąj ,A j )1,n ],[(a ji :Îąji ,A ji )n +1,p ] 0,n:m 1 ,n 1 :m 2 ,n 2 i y −1 Ip ,q :r:p [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 ;Îłj )1,n 1 ],[(c ′ ;Îł ′ )n 1 +1,p ′ ]; 0,1 ,[(e j ;E j )1,n 2 ],[(e ′′ ;E ′′ )n 2 +1,p ′′ ] ji ji ji ji 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 ′′ ],(1,1) ji ji ji ji i i đ?œ•2 đ?œ•đ?‘Ľđ?œ•đ?‘Ś

(9)

0,n:m 1 ,n 1 :m 2 ,n 2 x ′ ′′ [y ] i i :r:p i ′ ,q i ′ :r :p i ′′ ,q i ′′ :r

Ip ,q

[(a j :Îąj ,A j )1,n ],[(a ji :Îąji ,A ji )n +1,p i ] 0,n:m ,n :m 2 ,n 2 [x | :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 : 0,1 ,[(c j ;Îłj )1,n 1 ],[(c ′ ;Îł ′ )n 1 +1,p ′ ]; 0,1 ,[(e j ;E j )1,n 2 ],[(e ′′ ;E ′′ )n 2 +1,p ′′ ] ji ji ji ji i i ], :[(d j ;δj )1,m 1 ],[(d ′ ;δγ ′ )m 1 +1,q ′ ],(1,1);[(f j ;F j )1,m 2 ],[(f ′′ ;F ′′ )m 2 +1,q ′′ ],(1,1)

1 1 = (xy)−1 Ip ,q :r:p ,q

ji

ji

ji

i

ji

(10)

i

where | arg x | < ½ U'ď °, | arg y | < ½ V'ď °ď€Źď€ where U' and V' are given as follows respectively: q′ p′ pi qi m n1 i i đ?‘ˆ ′ = j=n+1 Îąji − j=1 βji + j=11 δj − j=m δ + j=1 Îłj − j=n Îł > 0, +1 ji ′ +1 ji ′ 1

đ?‘‰â€˛ = −

pi j=n+1 A ji

−

qi j=1 Bji

−

m2 j=1 Fj

−

1

q ′′ i F j=m 21 +1 ji ′′

+

n2 j=1 Ej

−

p ′′ i E j=n 2 +1 ji ′′

> 0,

References [1]. [2].

Goyal, Anil and Agrawal,R.D. Integral involving the product of I-function of two variables, Journal of M.A.C.T. Vol. 28 P 147155(1995). Rainville, E. D.: Special Functions, Macmillan, NewYork, 1960.

DOI: 10.9790/5736-1002012729

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