Gen. Math. Notes, Vol. 6, No. 1, September 2011, pp.1-14
ISSN 2219-7184; Copyright c ICSRS Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
Integrals Involving I-Function
U.K. Saha1 , L.K. Arora2 and B.K. Dutta3
1
Department of Mathematics
North Eastern Regional Institute of Science and Technology
Nirjuli-791109, Arunachal Pradesh, India
E-mail: uksahanerist@gmail.com
2
Department of Mathematics
North Eastern Regional Institute of Science and Technology
Nirjuli-791109, Arunachal Pradesh, India
E-mail: lkarora nerist@yahoo.com
3
Department of Mathematics
North Eastern Regional Institute of Science and Technology
Nirjuli-791 109, Arunachal Pradesh, India
E-mail: dutta.bk08@yahoo.com
(Received:11-4-11 /Accepted:10-7-11)
Abstract
In this paper, we have presented certain integrals involving product of the Ifunction with exponential function, Gauss’s hypergeometric function and Fox’s
H-function. The results derived here are basic in nature and may include a
number of known and new results as particular cases.
Keywords: Exponential function, hypergeometric function, H-function,
I-function, Mellin-Barnes type contour integral.
1
Introduction
The Gaussian hypergeometric function is of fundamental importance in the
theory of special functions. The importance of this function lies in the fact
that almost all of the commonly used functions of applicable mathematics,
mathematical physics, engineering and mathematical biology are expressible
as its special cases.
2
U.K. Saha et al.
The series
2 F1 (a, b; c; z)
=
∞
X
(a)n (b)n z n
n=0
(c)n
n!
,
(1)
where (a)n is the Pochhammer symbol defined by
(a)n =
a(a + 1) . . . (a + n − 1), n ∈ N
1,
n=0
(2)
is called the Gauss’s hypergeometric series after the famous German mathematician Carl Friedrich Gauss (1777-1855) who in the year 1812 introduced
this series. It is represented by the symbol 2 F1 (a, b; c; z) and is called the
Gauss’s hypergeometric function also.
In 1961, Charles Fox [2] introduced a function which is more general than
the Meijer’s G-function and this function is well known in the literature of special functions as Fox’s H-function or simply the H-function. This function is
defined and represented by means of the following Mellin-Barnes type contour
integral:
Z
1
(aj , αj )1,p
m,n
m,n
θ(s)z s ds,
(3)
H[z] = Hp,q [z] = Hp,q z
=
(bj , βj )1,q
2πi L
where, for convenience,
Qn
Qm
j=1 Γ(1 − aj + αj s)
j=1 Γ(bj − βj s)
Qp
,
θ(s) = Qq
j=n+1 Γ(aj − αj s)
j=m+1 Γ(1 − bj + βj s)
(4)
and L is a suitable contour of the Mellin-Barnes type which runs from −i∞
to +i∞, separating the poles of Γ(bj − βj s), j = 1, . . . , m from those of Γ(1 −
aj + αj s), j = 1, . . . , n. An empty product is interpreted as unity. The integers
m, n, p, q satisfy the inequalities 0 ≤ n ≤ p, 0 ≤ m ≤ q; the coefficients αj
(j = 1, . . . , p), βj (j = 1, . . . , q) are positive real numbers, and the complex
parameters aj (j = 1, . . . , p), bj (j = 1, . . . , q) are so constrained that no poles
of the integrand coincide. Owing to the popularity of the special functions,
those are defined in (1) and (3) (c.f. [4], [3] and [6]), details regarding these
are avoided.
The I-function, which is more general than the Fox’s H-function, defined
by V.P. Saxena [5], by means of the following Mellin-Barnes type contour
integral:
Z
1
(aj , αj )1,n ; (aji , αji )n+1,pi
m,n
m,n
I[z] = Ipi ,qi :r [z] = Ipi ,qi :r z
=
φ(ξ)z ξ dξ,
(bj , βj )1,m ; (bji , βji )m+1,qi
2πi L
(5)
3
Integrals Involving I-Function
where,
Qm
φ(ξ) =
Σri=1
nQ
Γ(bj − βj ξ)
Qn
Γ(1 − aj + αj ξ)
o,
Qpi
qi
Γ(1
−
b
+
β
ξ)
Γ(a
−
α
ξ)
ji
ji
ji
ji
j=m+1
j=n+1
j=1
j=1
(6)
pi , qi (i = 1, . . . , r), m, n are integers satisfying 0 ≤ n ≤ pi , 0 ≤ m ≤ qi ;
αj , βj , αji , βji are real and positive and aj , bj , aji , bji are complex numbers. L
is a suitable contour of the Mellin-Barnes type running from γ − iα to γ + iα
(γ is real) in the complex ξ-plane. Details regarding existence conditions and
various parametric restrictions of I-function, we may refer [5].
For r = 1, (5) reduces to the Fox’s H-function
(aj , αj )1,n ; (aji , αji )n+1,pi
(aj , αj )1,n ; (aj , αj )n+1,p
m,n
m,n
Ipi ,qi :1 z
= Hp,q z
(bj , βj )1,m ; (bji , βji )m+1,qi
(bj , βj )1,m ; (bj , βj )m+1,q
2
Required Results
We shall require the following results in the sequel:
The Mellin transform of the H-function follows from the definition (3) in
view of the well-known Mellin inversion theorem. We have
"
#
Z ∞
(a
,
α
)
j
j 1,p
m,n
xs−1 Hp,q
ax
dx = a−s θ(−s)
(b
,
β
)
j
j
1,q
0
Qn
Qm
j=1 Γ(1 − aj − αj s)
j=1 Γ(bj + βj s)
−s
Qp
,
(7)
= a Qq
j=n+1 Γ(aj + αj s)
j=m+1 Γ(1 − bj − βj s)
where,
A=
n
X
αj −
j=1
p
X
αj +
j=n+1
m
X
q
X
βj −
j=1
j=m+1
q
p
βj > 0,
X
X
1
| arg a| < Aπ, δ =
βj −
αj > 0
2
j=1
j=1
and
− min [Re(bj /βj )] < Re(s) < min [Re{(1 − aj )/αj }].
1≤j≤m
1≤j≤n
Lemma 2.1 From Rainville [4], we have
∞ X
∞
X
n=0 k=0
A(k, n) =
∞ X
n
X
n=0 k=0
A(k, n − k)
(8)
4
U.K. Saha et al.
3
Main Results
In this section, we have evaluated certain integrals involving product of the Ifunction with exponential function, Gauss’s hypergeometric function and Fox’s
H-function.
First Integral
Z
t
xρ−1 (t − x)σ−1 e−xz 2 F1 (α, β; γ; axζ (t − x)η )
0
m,n
µ
ν (aj , αj )1,n ; (aji , αji )n+1,pi
× Ipi ,qi :r yx (t − x)
dx
(bj , βj )1,m ; (bji , βji )m+1,qi
∞ X
n
X
z u−k (ζ+η−1)k+u
−zt ρ+σ−1
=e t
f (k)
t
(u − k)!
u=0 k=0
(1 − ρ − ζk, µ), (1 − σ − (η − 1)k − u, ν),
m,n+2
× Ipi +2,qi +1:r ytµ+ν
(bj , βj )1,m ; (bji , βji )m+1,qi ,
I1 ≡
(aj , αj )1,n ; (aji , αji )n+1,pi
,
(1 − ρ − σ − (ζ + η − 1)k − u, µ + ν)
(9)
where,
f (k) =
(α)k (β)k ak
,
(γ)k k!
provided
(i) µ ≥ 0, ν ≥ 0 (not both zero simultaneously)
(ii) ζ and η are non-negative integers such that ζ + η ≥ 1
1
(iii) Ai > 0,
∀i ∈ 1, . . .P
, r; where
i < 0; | arg
PB
Ppiy| < 2 Ai π,P
n
qi
Ai = j=1 αj − j=n+1 αji + m
β
−
j=1 j
j=m+1 βji ,
P
P
qi
pi
1
Bi = 2 (pi − qi ) + j=1 bji − j=1 aji
(iv) Re(ρ) + µ min1≤j≤m [Re(bj /βj )] > 0,
Re(σ) + ν min1≤j≤m [Re(bj /βj )] > 0.
Proof:
−zt
Z
t
xρ−1 (t − x)σ−1 e(t−x)z 2 F1 (α, β; γ; axζ (t − x)η )
0
µ
ν (aj , αj )1,n ; (aji , αji )n+1,pi
m,n
× Ipi ,qi :r yx (t − x)
dx
(bj , βj )1,m ; (bji , βji )m+1,qi
I1 ≡ e
(10)
5
Integrals Involving I-Function
P
(t−x)u z u
Now we replace e(t−x)z by ∞
and express the hypergeometric funcu=0
u!
tion and the I-function with the help of (1) and (5) respectively, to get
I1 = e
−zt
t
Z
x
ρ−1
σ−1
(t − x)
0
∞
∞
X
(t − x)u z u X (α)k (β)k ak xζk (t − x)ηk
u!
(γ)k
k!
u=0
k=0
Z
1
×
φ(ξ)y ξ xµξ (t − x)νξ dξdx
2πi L
Z t
∞ X
∞
X
(α)k (β)k ak xζk (t − x)ηk+u z u
ρ−1
σ−1
−zt
x (t − x)
=e
(γ)k
k!
u!
0
u=0 k=0
Z
1
×
φ(ξ)y ξ xµξ (t − x)νξ dξdx
2πi L
Now by the use of (8), the above result reduces to
−zt
Z
t
I1 = e
x
ρ−1
σ−1
(t − x)
0
1
×
2πi
Z
∞ X
n
X
(α)k (β)k ak xζk (t − x)ηk+u−k z u−k
(γ)k
k!
(u − k)!
u=0 k=0
φ(ξ)y ξ xµξ (t − x)νξ dξdx
L
Interchanging the order of integration and summation, we obtain
−zt
I1 = e
∞ X
n
X
u=0
t
z u−k 1
f (k)
(u − k)! 2πi
k=0
Z
×
x
ρ+ζk+µξ−1
Z
φ(ξ)y ξ
L
σ+(η−1)k+u+νξ−1
(t − x)
dx dξ,
0
where f (k) is given by (10).
On substituting x = ts in the inner x-integral, the above expression reduces
to
n
∞ X
X
Z
z u−k (ζ+η−1)k+u 1
I1 = e t
f (k)
t
φ(ξ)y ξ t(µ+ν)ξ
(u
−
k)!
2πi
L
u=0 k=0
Z 1
×
sρ+ζk+µξ−1 (1 − s)σ+(η−1)k+u+νξ−1 ds dξ
−zt ρ+σ−1
0
−zt ρ+σ−1
=e
t
n
∞ X
X
u=0
z u−k (ζ+η−1)k+u 1
f (k)
t
(u
−
k)!
2πi
k=0
Z
φ(ξ)
L
Γ(ρ + ζk + µξ)Γ(σ + (η − 1)k + u + νξ) ξ (µ+ν)ξ
×
y t
dξ
Γ(ρ + σ + (ζ + η − 1)k + u + (µ + ν)ξ)
6
U.K. Saha et al.
Finally, interpreting the contour integral by virtue of (5), we obtain
−zt ρ+σ−1
I1 = e
t
∞ X
n
X
f (k)
u=0 k=0
×
z u−k (ζ+η−1)k+u
t
(u − k)!
(1 − ρ − ζk, µ), (1 − σ − (η − 1)k − u, ν),
(bj , βj )1,m ; (bji , βji )m+1,qi ,
(aj , αj )1,n ; (aji , αji )n+1,pi
.
(1 − ρ − σ − (ζ + η − 1)k − u, µ + ν)
Ipm,n+2
i +2,qi +1:r
ytµ+ν
Second Integral
Z t
I2 ≡
xρ−1 (t − x)σ−1 e−xz 2 F1 (α, β; γ; axζ (t − x)η )
0
m,n
−µ
−ν (aj , αj )1,n ; (aji , αji )n+1,pi
dx
× Ipi ,qi :r yx (t − x)
(bj , βj )1,m ; (bji , βji )m+1,qi
∞ X
n
X
z u−k (ζ+η−1)k+u
= e−zt tρ+σ−1
f (k)
t
(u
−
k)!
u=0 k=0
(aj , αj )1,n ; (aji , αji )n+1,pi ,
m+2,n
× Ipi +1,qi +2:r yt−µ−ν
(ρ + ζk, µ), (σ + (η − 1)k + u, ν),
(ρ + σ + (ζ + η − 1)k + u, µ + ν)
,
(bj , βj )1,m ; (bji , βji )m+1,qi
(11)
provided
Re(ρ) − µ max [Re{(aj − 1)/αj }] > 0,
1≤j≤n
Re(σ) − ν max [Re{(aj − 1)/αj }] > 0,
1≤j≤n
along with the sets of conditions (i) to (iii) given with I1 and f (k) is given by
(10).
Third Integral
Z t
I3 ≡
xρ−1 (t − x)σ−1 e−xz 2 F1 (α, β; γ; axζ (t − x)η )
0
µ
−ν (aj , αj )1,n ; (aji , αji )n+1,pi
m,n
dx
× Ipi ,qi :r yx (t − x)
(bj , βj )1,m ; (bji , βji )m+1,qi
∞ X
n
X
z u−k (ζ+η−1)k+u
−zt ρ+σ−1
t
=e t
f (k)
(u − k)!
u=0 k=0
(1 − ρ − ζk, µ),
m+1,n+1
× Ipi +1,qi +2:r ytµ−ν
(σ + (η − 1)k + u, ν), (bj , βj )1,m ;
7
Integrals Involving I-Function
(aj , αj )1,n ; (aji , αji )n+1,pi
,
(bji , βji )m+1,qi , (1 − ρ − σ − (ζ + η − 1)k − u, µ − ν)
(12)
provided µ > 0, ν ≥ 0 such that µ − ν ≥ 0,
Re(ρ) + µ min [Re(bj /βj )] > 0,
1≤j≤m
Re(σ) − ν max [Re{(aj − 1)/αj }] > 0,
1≤j≤n
along with the sets of conditions (i) to (iii) given with I1 and f (k) is given by
(10).
Fourth Integral
Z t
xρ−1 (t − x)σ−1 e−xz 2 F1 (α, β; γ; axζ (t − x)η )
I4 ≡
0
m,n
µ
−ν (aj , αj )1,n ; (aji , αji )n+1,pi
× Ipi ,qi :r yx (t − x)
dx
(bj , βj )1,m ; (bji , βji )m+1,qi
∞ X
n
X
z u−k (ζ+η−1)k+u
t
= e−zt tρ+σ−1
f (k)
(u
−
k)!
u=0 k=0
(1 − ρ − ζk, µ), (aj , αj )1,n ;
m+1,n+1
× Ipi +2,qi +1:r ytµ−ν
(σ + (η − 1)k + u, ν),
(aji , αji )n+1,pi , (ρ + σ + (ζ + η − 1)k + u, ν − µ)
,
(bj , βj )1,m ; (bji , βji )m+1,qi
(13)
provided µ ≥ 0, ν > 0 such that ν − µ ≥ 0,
Re(ρ) − µ max [Re{(aj − 1)/αj }] > 0,
1≤j≤n
Re(σ) + ν min [Re(bj /βj )] > 0,
1≤j≤m
along with the sets of conditions (i) to (iii) given with I1 and f (k) is given by
(10).
Fifth Integral
Z t
I5 ≡
xρ−1 (t − x)σ−1 e−xz 2 F1 (α, β; γ; axζ (t − x)η )
0
m,n
−µ
ν (aj , αj )1,n ; (aji , αji )n+1,pi
dx
× Ipi ,qi :r yx (t − x)
(bj , βj )1,m ; (bji , βji )m+1,qi
∞ X
n
X
z u−k (ζ+η−1)k+u
−zt ρ+σ−1
=e t
f (k)
t
(u − k)!
u=0 k=0
(1 − σ − (η − 1)k − u, ν), (aj , αj )1,n ;
m+1,n+1
× Ipi +2,qi +1:r yt−µ+ν
(ρ + ζk, µ),
8
U.K. Saha et al.
(aji , αji )n+1,pi , (ρ + σ + (ζ + η − 1)k + u, µ − ν)
,
(bj , βj )1,m ; (bji , βji )m+1,qi
(14)
provided µ > 0, ν ≥ 0 such that µ − ν ≥ 0,
Re(ρ) + µ min [Re(bj /βj )] > 0,
1≤j≤m
Re(σ) − ν max [Re{(aj − 1)/αj }] > 0,
1≤j≤n
along with the sets of conditions (i) to (iii) given with I1 and f (k) is given by
(10).
Sixth Integral
Z
t
xρ−1 (t − x)σ−1 e−xz 2 F1 (α, β; γ; axζ (t − x)η )
0
m,n
−µ
ν (aj , αj )1,n ; (aji , αji )n+1,pi
× Ipi ,qi :r yx (t − x)
dx
(bj , βj )1,m ; (bji , βji )m+1,qi
∞ X
n
X
z u−k (ζ+η−1)k+u
−zt ρ+σ−1
t
=e t
f (k)
(u
−
k)!
u=0 k=0
(1 − σ − (η − 1)k − u, ν), (aj , αj )1,n ;
m+1,n+1
× Ipi +1,qi +2:r yt−µ+ν
(ρ + ζk, µ), (bj , βj )1,m ; (bji , βji )m+1,qi ,
I6 ≡
(aji , αji )n+1,pi
,
(1 − ρ − σ − (ζ + η − 1)k − u, ν − µ)
(15)
provided µ ≥ 0, ν > 0 such that ν − µ ≥ 0,
Re(ρ) − µ max [Re{(aj − 1)/αj }] > 0,
1≤j≤n
Re(σ) + ν min [Re(bj /βj )] > 0,
1≤j≤m
along with the sets of conditions (i) to (iii) given with I1 and f (k) is given by
(10).
The integrals (11) to (15) can be proved on lines similar to those of integral
(9).
9
Integrals Involving I-Function
Seventh Integral
Z ∞
σ (aj , αj )1,n1 ; (aji , αji )n1 +1,pi
η−1 ax
ρ m1 ,n1
I7 ≡
x e 2 F1 (α, β; γ; ax )Ipi ,qi :r zx
(bj , βj )1,m1 ; (bji , βji )m1 +1,qi
0
(cj , γj )1,n ; (cj , γj )n+1,p
m,n
wx
dx
× Hp,q
(dj , δj )1,m ; (dj , δj )m+1,q
∞ X
n
X
au−k
−η
=w
f (k)
w−(ρ−1)k−u
(u
−
k)!
u=0 k=0
(aj , αj )1,n1 , (1 − dj − (η + (ρ − 1)k + u)δj , σδj )1,m ;
m1 +n,n1 +m
× Ipi +q,qi +p:r zw−σ
(bj , βj )1,m1 , (1 − cj − (η + (ρ − 1)k + u)γj , σγj )1,n ;
(aji , αji )n1 +1,pi , (1 − dj − (η + (ρ − 1)k + u)δj , σδj )m+1,q
(bji , βji )m1 +1,qi , (1 − cj − (η + (ρ − 1)k + u)γj , σγj )n+1,p
,
(16)
where,
(α)k (β)k ak
,
f (k) =
(γ)k k!
(17)
provided
(i) λ > 0, | arg z| < 12 πλ
(ii) λ ≥ 0, | arg z| ≤ 12 πλ, Re(µ + 1) < 0
(iii) λ1 > 0, | arg w| < 12 πλ1
(iv) λ1 ≥ 0, | arg w| ≤ 12 πλ1 , Re(µ1 + 1) < 0
(v) σ > 0, −σ min1≤j≤m1 [Re(bj /βj )] − min1≤j≤m [Re(dj /δj )] < Re(η) < σ <
min1≤j≤n1 [Re{(1 − aj )/αj }] + min1≤j≤n [Re{(1 − cj )/γj }]
and where,
λ=
µ=
λ1 =
n1
X
αj +
m1
X
j=1
j=1
m1
X
n1
X
j=1
m
X
j=1
bj −
δj +
j=1
n
X
pi
X
"
βj − max
1≤i≤r
j=n1 +1
1≤i≤r
X
γj −
j=1
q
X
j=m+1
p
aji −
j=n1 +1
q
X
X
1
µ1 = (p − q) +
dj −
cj
2
j=1
j=1
qi
X
j=m1 +1
p
δj −
X
j=n+1
#
βji
j=m1 +1
pi
"
aj − min
αji +
qi
X
γj
bji +
pi q i
−
2
2
#
10
U.K. Saha et al.
P∞ au xu
Proof: We replace eax by
and express the hypergeometric
u=0 u!
function and the I-function with the help of (1) and (5) respectively, to obtain
Z ∞
Z
∞
∞
X
au xu X (α)k (β)k ak xρk 1
η−1
x
I7 =
φ(ξ)z ξ xσξ
u!
(γ)
k!
2πi
k
0
L
u=0
k=0
(cj , γj )1,n ; (cj , γj )n+1,p
m,n
× Hp,q
wx
dξdx
(dj , δj )1,m ; (dj , δj )m+1,q
Z ∞
Z
∞ X
∞
X
(α)k (β)k ak xρk+u au 1
η−1
x
=
φ(ξ)z ξ xσξ
(γ)
k!
u!
2πi
k
0
L
u=0 k=0
(cj , γj )1,n ; (cj , γj )n+1,p
m,n
× Hp,q wx
dξdx
(dj , δj )1,m ; (dj , δj )m+1,q
Now by the use of (8), the above result reduces to
Z
Z ∞
∞ X
n
X
(α)k (β)k ak xρk+u−k au−k 1
η−1
φ(ξ)z ξ xσξ
x
I7 =
(γ)
k!
(u
−
k)!
2πi
k
L
0
u=0 k=0
(cj , γj )1,n ; (cj , γj )n+1,p
m,n
× Hp,q
wx
dξdx
(dj , δj )1,m ; (dj , δj )m+1,q
Interchanging the order of integration and summation, we obtain
Z ∞
Z
∞ X
n
X
au−k 1
ξ
f (k)
I7 =
φ(ξ)z
xη+(ρ−1)k+u+σξ−1
(u
−
k)!
2πi
L
0
u=0 k=0
(cj , γj )1,n ; (cj , γj )n+1,p
m,n
× Hp,q wx
dx dξ,
(dj , δj )1,m ; (dj , δj )m+1,q
where f (k) is given by (17).
Now we use the Mellin transform of H-function by virtue of (7), so that
Z
∞ X
n
X
au−k 1
φ(ξ)z ξ w−(η+(ρ−1)k+u+σξ)
I7 =
f (k)
(u
−
k)!
2πi
L
u=0 k=0
Qm
j=1 Γ(dj + δj (η + (ρ − 1)k + u + σξ))
× Qq
Γ(1 − dj − δj (η + (ρ − 1)k + u + σξ))
Qj=m+1
n
j=1 Γ(1 − cj − γj (η + (ρ − 1)k + u + σξ))
× Qp
dξ
j=n+1 Γ(cj + γj (η + (ρ − 1)k + u + σξ))
Z
∞ X
n
X
au−k
−η
−(ρ−1)k−u 1
= w
f (k)
w
φ(ξ)
(u − k)!
2πi L
u=0 k=0
Qm
j=1 Γ(dj + (η + (ρ − 1)k + u)δj + σδj ξ))
× Qq
Γ(1 − dj − (η + (ρ − 1)k + u)δj − σδj ξ))
Qj=m+1
n
j=1 Γ(1 − cj − (η + (ρ − 1)k + u)γj − σγj ξ)) ξ −σξ
× Qp
z w dξ
j=n+1 Γ(cj + (η + (ρ − 1)k + u)γj + σγj ξ))
11
Integrals Involving I-Function
Finally, interpreting the contour integral by virtue of (5), we obtain
I7 =w
−η
∞ X
n
X
f (k)
u=0 k=0
×
1 +n,n1 +m
Ipmi +q,q
i +p:r
au−k
w−(ρ−1)k−u
(u − k)!
(aj , αj )1,n1 , (1 − dj − (η + (ρ − 1)k + u)δj , σδj )1,m ;
zw−σ
(bj , βj )1,m1 , (1 − cj − (η + (ρ − 1)k + u)γj , σγj )1,n ;
(aji , αji )n1 +1,pi , (1 − dj − (η + (ρ − 1)k + u)δj , σδj )m+1,q
(bji , βji )m1 +1,qi , (1 − cj − (η + (ρ − 1)k + u)γj , σγj )n+1,p
4
.
Particular Cases
Putting r = 1, t = 1 and η = 0 in (9), (11), (12), (13), (14) and (15) the
following known as well as new results may be realised:
(i) Integral (9) leads to the known result [1, p. 246, eq. (2.2)]:
Z 1
ρ−1
σ−1 −xz
ζ
m,n
µ
ν (aj , αj )1,p
x (1 − x) e 2 F1 (α, β; γ; ax )Hp,q yx (1 − x)
dx
(bj , βj )1,q
0
∞ X
n
X
z u−k
(1 − ρ − ζk, µ),
m,n+2
−z
=e
f (k)
H
y
(bj , βj )1,q ,
(u − k)! p+2,q+1
u=0 k=0
(1 − σ + k − u, ν), (aj , αj )1,p
,
(1 − ρ − σ − (ζ − 1)k − u, µ + ν)
(18)
where,
f (k) =
(α)k (β)k ak
.
(γ)k k!
(ii) Integral (11) leads to the another known result [1, p. 248, eq. (3.1)]:
Z 1
ζ
m,n
−µ
−ν (aj , αj )1,p
ρ−1
σ−1 −xz
x (1 − x) e 2 F1 (α, β; γ; ax )Hp,q yx (1 − x)
dx
(bj , βj )1,q
0
∞ X
n
X
z u−k
(aj , αj )1,p ,
m+2,n
−z
f (k)
=e
Hp+1,q+2 y
(ρ + ζk, µ),
(u − k)!
u=0 k=0
(ρ + σ + (ζ − 1)k + u, µ + ν)
,
(σ − k + u, ν), (bj , βj )1,q
where,
f (k) =
(α)k (β)k ak
.
(γ)k k!
(19)
12
U.K. Saha et al.
(iii) Integral (12) reduces to the known result [1, p. 248, eq. (3.2)]:
Z
1
x
ρ−1
(1 − x)
σ−1 −xz
e
2 F1 (α, β; γ; ax
ζ
m,n
)Hp,q
µ
−ν
yx (1 − x)
0
=e
−z
(aj , αj )1,p
(bj , βj )1,q
dx
∞ X
n
X
u=0
z u−k
(1 − ρ − ζk, µ),
m+1,n+1
f (k)
Hp+1,q+2 y
(σ − k + u, ν), (bj , βj )1,q ,
(u − k)!
k=0
(aj , αj )1,p
,
(1 − ρ − σ − (ζ − 1)k − u, µ − ν)
(20)
where,
f (k) =
(α)k (β)k ak
.
(γ)k k!
(iv) Integral (13) leads to the known result [1, p. 249, eq. (3.3)]:
Z
1
x
ρ−1
(1 − x)
σ−1 −xz
e
2 F1 (α, β; γ; ax
ζ
m,n
)Hp,q
µ
−ν
yx (1 − x)
0
=e
−z
(aj , αj )1,p
(bj , βj )1,q
dx
∞ X
n
X
u=0
z u−k
(1 − ρ − ζk, µ), (aj , αj )1,p ,
m+1,n+1
Hp+2,q+1 y
f (k)
(σ
− k + u, ν),
(u − k)!
k=0
(ρ + σ + (ζ − 1)k + u, ν − µ)
,
(bj , βj )1,q
(21)
where,
f (k) =
(α)k (β)k ak
.
(γ)k k!
(v) Integral (14) reduces to the result:
Z
1
x
ρ−1
(1 − x)
σ−1 −xz
e
2 F1 (α, β; γ; ax
ζ
m,n
)Hp,q
yx
0
=e
−z
−µ
ν
(1 − x)
(aj , αj )1,p
(bj , βj )1,q
dx
∞ X
n
X
u=0
z u−k
(1 − σ + k − u, ν), (aj , αj )1,p ,
m+1,n+1
f (k)
Hp+2,q+1 y
(ρ + ζk, µ),
(u − k)!
k=0
(ρ + σ + (ζ − 1)k + u, µ − ν)
,
(bj , βj )1,q
where,
f (k) =
(α)k (β)k ak
.
(γ)k k!
(22)
13
Integrals Involving I-Function
(vi) Integral (15) leads to the result:
Z
1
x
ρ−1
(1 − x)
σ−1 −xz
e
2 F1 (α, β; γ; ax
ζ
m,n
)Hp,q
−µ
yx
ν
(1 − x)
0
=e
−z
(aj , αj )1,p
(bj , βj )1,q
dx
∞ X
n
X
u=0
z u−k
(1 − σ + k − u, ν),
m+1,n+1
f (k)
Hp+1,q+2 y
(ρ + ζk, µ), (bj , βj )1,q ,
(u − k)!
k=0
(aj , αj )1,p
,
(1 − ρ − σ − (ζ − 1)k − u, ν − µ)
(23)
where,
f (k) =
(α)k (β)k ak
.
(γ)k k!
(vii) Putting a = 0 in (16), the exponential function eax and the hypergeometric function reduces to unity and consequently it leads to a result by V.P.
Saxena [5, p. 66, eq. (4.5.1)]:
∞
(aj , αj )1,n ; (aji , αji )n+1,pi
zx
(bj , βj )1,m ; (bji , βji )m+1,qi
0
(cj , γj )1,n ; (cj , γj )n+1,p
m,n
× Hp,q wx
dx
(dj , δj )1,m ; (dj , δj )m+1,q
(aj , αj )1,n1 , (1 − dj − ηδj , σδj )1,m ;
−η m1 +n,n1 +m
=w Ipi +q,qi +p:r zw−σ
(bj , βj )1,m1 , (1 − cj − ηγj , σγj )1,n ;
Z
1
xη−1 Ipmi ,q1 ,n
i :r
σ
(aji , αji )n1 +1,pi , (1 − dj − ηδj , σδj )m+1,q
(bji , βji )m1 +1,qi , (1 − cj − ηγj , σγj )n+1,p
5
.
(24)
Conclusion
The I-function, presented in this paper, is quite basic in nature. Therefore,
on specializing the parameters of this function, we may obtain various other
special functions such as Fox’s H-function, Meijer’s G-function, Wright’s generalized Bessel function, Wright’s generalized hypergeometric function, MacRobert’s E-function, generalized hypergeometric function, Bessel function of
first kind, modified Bessel function, Whittaker function, exponential function,
binomial function etc. as its special cases, and therefore, various unified integral presentations can be obtained as special cases of our results.
14
U.K. Saha et al.
References
[1] L.K. Arora and U.K. Saha, Integrals involving hypergeometric function
and H-function, J. Indian Acad. Math., 32(1) (2010), 243-249.
[2] C. Fox, The G- and H-functions as symmetrical Fourier kernels, Trans.
Amer. Math. Soc., 98(1961), 395-429.
[3] A.M. Mathai and R.K. Saxena, The H-Function with Applications in
Statistics and Other Disciplines, Wiley Eastern Limited, New Delhi, Bangalore, Bombay, (1978).
[4] E.D. Rainville, Special Functions, Chelsea Publication Company, Bronx,
New York, (1971).
[5] V.P. Saxena, The I-Function, Anamaya Publishers, New Delhi, (2008).
[6] H.M. Srivastava, K.C. Gupta and S.P. Goyal, The H-Functions of One
and Two Variables with Applications, South Asian Publishers, New Delhi,
Madras, (1982).