The aim of this research paper is to establish some linear generating relations involving I-function of two variables.

1. 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
DOI: 10.9790/5728-1301012325 www.iosrjournals.org 23 | Page
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:
I[y
x
] = Ipi,qi:r:pi′ ,qi′ :r′ :pi′′ ,qi′′ :r′′
0,n:m1,n1:m2,n2
[y
x
|[(bji :βji ,Bji )1,qi
]
[(aj:αj,Aj)1,n ],[(aji :αji ,Aji )n+1,pi
]
]:[(dj ;δj )1,m 1],[(dji′ ;δγji′ )m 1+1,q
i′ ];[(fj;Fj)1,m 2],[(fji′′ ;Fji′′ )m 2+1,q
i′′ ]
:[(cj;γj )1,n1],[(cji′ ;γji′ )n1+1,p
i′ ];[(ej;Ej)1,n2],[(eji′′ ;Eji′′ )n2+1,p
i′′ ]
=
1
(2πω )2 ϕ1 ξ, η θ2 ξ θ3(η)L2L1
xξ
yη
dξdη, (1)
where
ϕ1 ξ, η =
Γ(1−aj+αjξ+Ajη)n
j=1
[r
i=1 Γ(aji −αji ξ−Aji η)
pi
j=n+1
Γ(1−bji +βji ξ+Bji η)
qi
j=1
,
θ2 ξ =
Γ(dj−δj ξ)
m 1
j=1 Γ(1−cj +γjξ)
n1
j=1
[ Γ(1−dji′ +δji′ ξ)
q
i′
j=m 1+1
Γ(cji′ −γji′ ξ)
p
i′
j=n1+1
r′
i′ =1
,
θ3 η =
Γ(fj−Fj η)
m 2
j=1 Γ(1−ej+Ejη)
n2
j=1
[ Γ(1−fji′′ +Fji′′ η)
q
i′′
j=m 2+1
Γ(eji′′ −Eji′′ η)
p
i′′
j=n2+1
r′′
i′′ =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 n 0, pi´ n1 0, pi´´ n2 0, qi >0, qi´  0, qi´´  0, (i = 1, …, r; i´
= 1, …, r´; i´´ = 1, …, r´´; k = 1, 2) also all the A’s, ’s, B’s, ’s, ’s, ’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 plane and runs
from –  to + , with loops, if necessary, to ensure that the poles of djj) (j = 1, ..........., m1) lie to the
right, and the poles of cj  j) (j = 1, ..., n1), aj + j+ Aj) (j = 1, ..., n) to the left of the contour.
The contour L2 is in the plane and runs from –  to + , with loops, if necessary, to ensure that
the poles of fj  Fj) (j=1,....., n2) lie to the right, and the poles of ej  Ej) (j = 1, ..., m2), aj +
j+ Aj) (j = 1, ..., n) to the left of the contour. Also
𝑅′
= αji
pi
j=1 + γji′
pi′
j=1
− βji
qi
j=1 − δji′
qi′
j=1
< 0,
𝑆′
= Aji
pi
j=1 + Eji′′
pi′′
j=1
− Bji
qi
j=1 − Fδji′
qi′′
j=1
< 0,
𝑈 = αji
pi
j=n+1 − βji
qi
j=1 + δj
m1
j=1 − δji′
qi′
j=m1+1
+ γj
n1
j=1 − γji′
pi′
j=n1+1
> 0,
(2)
𝑉 = − Aji
pi
j=n+1 − Bji
qi
j=1 − Fj
m2
j=1 − Fji′′
qi′′
j=m21+1
+ Ej
n2
j=1 − Eji′′
pi′′
j=n2+1
> 0,
(3)
and |arg x| < ½ U, |arg y| < ½ V

2. On Some New Linear Generating Relations Involving I-Function of Two Variables
DOI: 10.9790/5728-1301012325 www.iosrjournals.org 24 | Page
II. Linear Generating Relations
In this section we establish the following linear generating relations:
t𝑙
𝑙!
∞
𝑙=0
Ipi,qi:r ∶pi′ +1,qi′ :r′ ∶pi′′ ,qi′′ :r′′
0, n ∶m1,n1 ∶m2,n2
|y
x
]……….,……..:……….,……..:………,……..
…,…:……….., λ−𝑙; α :………,……..
= (1 + t) ()
Ipi,qi:r ∶pi′ +1,qi′ :r′ ∶pi′′ ,qi′′ :r′′
0, n ∶m1,n1 ∶m2,n2
|y
x(1+𝑡)α
]……….,……..:……….,……..:………,……..
…,…:……….., λ;α :………,……..
, (4)
|arg x| < ½ U, |arg y| < ½ V, where U and V is given in (2) and (3) respectively;
t𝑙
𝑙!
∞
𝑙=0
Ipi,qi:r ∶pi′ +1,qi′ :r′ ∶pi′′ ,qi′′ :r′′
0, n ∶m1,n1+1 ∶m2,n2
|y
x
]……….,……..:……….,……..:………,……..
…,…: −λ−𝑙; α ,…….:………,……..
= (1  t) ()
Ipi,qi:r ∶pi′ +1,qi′ :r′ ∶pi′′ ,qi′′ :r′′
0, n ∶m1,n1+1 ∶m2,n2
|y
x(1−𝑡)−α
]……….,……..:……….,……..:………,……..
…,…: −λ;α ,…….:………,……..
, (5)
|arg x| < ½ U, |arg y| < ½ V, where U and V is given in (2) and (3) respectively.
Proof:
To prove (4), consider
 =
t𝑙
𝑙!
∞
𝑙=0
Ipi,qi:r ∶pi′ +1,qi′ :r′ ∶pi′′ ,qi′′ :r′′
0, n ∶m1,n1 ∶m2,n2
|y
x
]……….,……..:……….,……..:………,……..
…,…:……….., λ−𝑙; α :………,……..
On expressing I-function in contour integral form as given in (1), we get
 =
t𝑙
𝑙!
∞
𝑙=0
[
1
(2πω)2
ϕ1(ξ, η)
L2L1
θ2 ξ θ3 η

1
Γ{λ−𝑙−αξ}
xξ
yη
dξdη]
=
(−t)𝑙
𝑙!
∞
𝑙=0
[
1
(2πω)2
ϕ1(ξ, η)
L2L1
θ2 ξ θ3 η

{1−λ−αξ}𝑙
Γ{λ−αξ}
xξ
yη
dξdη].
On changing the order of summation and integration, we have
 =
1
(2πω)2
ϕ1(ξ, η)
L2L1
θ2 ξ θ3 η xξ
yη

1
Γ{λ−αξ}
 [
(−t)𝑙
𝑙!
∞
𝑙=0 {1 − λ − αξ}𝑙] dξdη
= (1 + t)λ−1 1
(2πω)2 ϕ1(ξ, η)L2L1
θ2 ξ θ3 η xξ
yη (1+t)−αξ
Γ{λ−αξ}
dξdη

3. On Some New Linear Generating Relations Involving I-Function of Two Variables
DOI: 10.9790/5728-1301012325 www.iosrjournals.org 25 | Page
which in view of (1), provides (4).
Proceeding on similar lines as above, the results (5) can be derived.
III. Particular Cases
I. On specializing the parameters in main formulae, we get following generating relations in terms of I-function
of one variable, which are the results given by Khare [1, p.21-23, (2.1) and (2.2)]:
t𝑙
𝑙!
∞
𝑙=0
Ipi+1,qi:r
m,n
z ...,…
…, λ−𝑙,α
]
= (1 + t) ()
Ipi+1,qi:r
m,n
[z 1 + t −α
|…,….
…,,(λ,α)
], (6)
provided that |argz| < ½ B, where B is given by 𝐵 = 𝛼𝑗
𝑛
𝑗=1 − 𝛼𝑗𝑖
𝑝 𝑖
𝑗=𝑛+1 + 𝛽𝑗
𝑚
𝑗 =1 − 𝛽𝑗𝑖
𝑞 𝑖
𝑗 =𝑚+1 > 0;
t𝑙
𝑙!
∞
𝑙=0
Ipi+1,qi:r
m,n+1
[z|……,
−λ−𝑙,α ,…
]
= (1  t) (1 + )
Ipi+1,qi:r
m,n+1
[z 1 − t −α
|……,.
−λ,α ,…
], (7)
provided that |argz| < ½ B, where B is given by 𝐵 = 𝛼𝑗
𝑛
𝑗=1 − 𝛼𝑗𝑖
𝑝 𝑖
𝑗=𝑛+1 + 𝛽𝑗
𝑚
𝑗 =1 − 𝛽𝑗𝑖
𝑞 𝑖
𝑗 =𝑚+1 > 0.
II. On choosing r = 1 in (6) and (7), we get following generating relations in terms of H-function of one
variable, which are the results given by Shrivastava & Shrivastava [3, p.65, (2.1) and (2.2)]:

(t)𝑙
𝑙!
∞
𝑙=0
Hp+1,q
m,n
[z| (bj,βj )1,q
(aj,αj)1,p ,(λ−𝑙,α)
]
= (1 + t) ()
Hp+1,q
m,n
[z 1 + t −α
| (bj,βj)1,q
(aj,αj)1,p ,(λ,α)
], (8)
provided that |argz| < ½ A, where A is given by 𝐴 = 𝛼𝑗
𝑛
𝑗 =1 − 𝛼𝑗
𝑝
𝑗=𝑛+1 + 𝛽𝑗
𝑚
𝑗=1 − 𝛽𝑗
𝑞
𝑗=𝑚+1 > 0;
(t)𝑙
𝑙!
∞
𝑙=0
Hp+1,q
m,n+1
[z| (bj,βj)1,q
−λ−𝑙,α ,(aj,αj)1,p
]
= (1  t) (1 + )
Hp+1,q
m,n+1
[z 1 − t −α
| (bj,βj )1,q
−λ,α ,(aj,αj)1,p
], (9)
provided that |argz| < ½ A, where A is given by 𝐴 = 𝛼𝑗
𝑛
𝑗 =1 − 𝛼𝑗
𝑝
𝑗=𝑛+1 + 𝛽𝑗
𝑚
𝑗=1 − 𝛽𝑗
𝑞
𝑗=𝑚+1 > 0.
References
[1]. Khare, Sandeep: An Advanced Study of I-Function of one Variable with Some Applications, Ph.D. Thesis, A.P.S. University, Rewa
(M.P.), 2009.
[2]. Sharma C. K. and Mishra, P. L.: On the I-function of two variables and its certain properties, ACI, 17 (1991), 1-4.
[3]. Shrivastava, Shweta and Shrivastava, B. M. L.: Some new generating relations and identities for H-function, Vijnana Parishad
Anusandhan Patrika, Vol. 49, No.1, January, 2006, p. 63-77.