DOI : 10.17577/IJERTCONV4IS12008
- Open Access
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- Authors : Rajeev Kumar Gupta, Devendra Singh Rawat, Dinesh Kumar
- Paper ID : IJERTCONV4IS12008
- Volume & Issue : ETRASECT – 2016 (Volume 4 – Issue 12)
- Published (First Online): 24-04-2018
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Certain Integrals of Generalized Hypergeometric and Confluent Hypergeometric Functions
Rajeev Kumar Gupta, Devendra Singh Rawat and Dinesh Kumar
Department of Mathematics & Statistics Jai Narain Vyas University,
Jodhpur – 342005, India
p
p
Abstract:- In this paper, we aim at establishing certain finite integral formulas for the generalized Gauss hypergeometric and confluent hypergeometric functions. Fur- thermore, the F (,)(a, b; c; z)-function occurring in each of our main results can
be reduced, under various special cases, to such simpler functions as the classi- cal Gauss hypergeometric function 2F1, Gauss confluent hypergeometric function
p
p
(,)(b; c; z) function and generalized hypergeometric function
pFq . A specimen
of some of these interesting applications of our main integral formulas are pre- sented briefly.
2010 Mathematics Subject Classification: 33B20, 33C05, 33C20, 33C99 . Keywords: Generalized Beta functions, Generalized Gamma functions, Generalized Gauss hypergeometric functions, Confluent hypergeometric functions.
1 INTRODUCTION AND PRELIMINARIES
In many areas of applied mathematics, various types of special functions become es- sential tools for scientists and engineers. The continuous development of mathematical physics, probability theory and other areas has led to new classes of special functions and their extensions and generalizations (see, for details, [17] and the references cited therein; see also [16, 18, 19]).
A lot of research work has recently come up on the study and development of the func- tions, which are more general than the Beta type function (x, y), popularly known as generalized Beta type functions. These functions, as a part of the theory of confluent hypergeometric functions, are important special functions and their closely related ones are widely used in physics and engineering. Moreover, generalized Beta functions [2, 3] have played a pivotal role in the advancement of further research and have proved to be exemplary in nature. The Eulers gamma function (z) is one of the most fundamental special functions, because of its important role in various fields in the mathematical, physical, engineering and statistical sciences. Various generalizations of the gamma function can be found in the literature [1, 5, 7, 9, 21].
t )
t )
The following extension of the gamma function is introduced by Chaudhry and Zubair [1]:
(z) = ( (0 t
p
p
z1
exp (t p dt, (p) > 0; z C
. (1.1)
(z), (z) > 0
The extension of Eulers beta function is considered by Chaudhry et al. [2] in the following form:
r
r
1
p (x, y) =
0
tx1 (1 t)y exp ( p \ dt, (p) > 0, (x) > 0, (y) > 0.
t(1 t)
t(1 t)
1
1
(1.2)
Chaudhry et al. [3] used p (x, y) to extend the hypergeometric function, known as the extended Gauss hypergeometric function, as follows:
n
n
·
·
n!
n!
p (b+n, cb) (z)
Fp (a, b; c; z) = (a)n
n=0
(b, cb)
, p 0, (c) > (b) > 0, (1.3)
where (a)n denotes the Pochhammer symbol defined as
(a) =
(a + n) 1, n= 0; a C/ {0}
= (
= (
n (a)
a (a+1) (a+2) . . . (a+n1) , n N,a C.
The integral representation of Eulers type function is
r
r
1 1
0
0
Fp (a, b; c; z) = (b,cb)
tb1 (1 t)c (1 zt) exp p dt, (1.4)
b1 a
b1 a
( \
( \
t (1t)
| |
| |
where p 0 and arg(1 z) < < p; (c) > (b) > 0.
Also, the extended confluent hypergeometric function is defined as
n
n
·
·
n!
n!
p (b+n, cb) (z)
p (b; c; z) =
n=0
(b, cb)
, p 0, (c) > (b) >0. (1.5)
The transformation formulas, recurrence relations, summation and asymptotic formu- las, differentiation properties, the Mellin transforms and some new representations of these extended functions can be found in many earlier work [3, 10, 12, 22, 19].
The generalized Eulers gamma function is defined in [12] as
p
p
r
r
p
p
(,) (x) =
0
tx1
1F1
(; ; t t dt, () > 0, () > 0, (p) > 0, (x) > 0.
(1.6)
Recently, O¨ zergin [11] introduced and studied some fundamental properties and char-
p
p
acteristics of the generalized Beta type function (,)(x, y) in their Ph.D. Thesis and
1
1
defined by (see, e.g., [11, p.32]):
r
r
1
p(,) (x, y) =
0
tx1(1t)y
1F1
; ; p dt, (1.7)
( \
( \
t (1t)
(,)
(,)
with (p) 0, min ( (x) , (y) , () , ()) > 0 and 0 (x, y) = (x, y) , where
1
1
(x, y) is a well known Eulers Beta function defined by
r
r
1
(x, y) =
0
tx1(1t)y dt, (((x)) > 0, ((y)) > 0) . (1.8)
p
p
Similarly, by appealing to ,(x, y),
O¨ zergin et al. introduced and investigated a
further extension of the following potentially useful generalized Gauss hypergeometric functions defined as follows (see, e.g., [12, p.4606, Sec.3]; see also [11, p.39, Ch.4]):
Fp(,) (a, b; c; z) = · (a)
p(,) (b + n, c b) zn
, (|z| < 1) , (1.9)
and
n
n=0
·
·
(b, c b) n!
(,,;p)
p(,) (b+n, cb) zn
1F1 (b; c; z) =
n=0
(b, cb)
, ( z < 1) , (1.10)
| |
| |
n!
corresponding integral representations are given by [12]:
F (,) (a, b; c; z) = 1 r
p
p
1
b1
b1
tb1(1 t)c
F (; ; p \ (1 zt)a dt,
1 1
1 1
(b, c b) 0
t (1 t)
(1.11)
| |
| |
for (p) 0, and arg (1 z) < < p; (c) > (b) > 0.
It is obvious to see that [3]:
F (,) (a, b; c; z) = Fp (a, b; c; z) , F (,) (a, b; c; z) = F (a, b; c; z) ;
p 0 2 1
1
1
1
1
F (,;p) (b; c; z) =
1Fp (b; c; z) = p
(b; c; z) ,
F (,;0) (b; c; z) =
1F1
(b; c; z) . (1.12)
1
1
1
1
1
1
where the 2F1(.) is a special case of the well-known generalized hypergeometric series
pFq (.) defined by (see, e.g., [19, Sec.1.5]; see also [20]).
( )1
( )1
· · · ( )q
· · · ( )q
n!
n!
n
p
p
q
q
1, …, q ;
1, …, q ;
F 1, …, p;
zl = · (1)n · · · (p)n z =
n=0
n
n
n=0
n
n
p
p
q
q
1
1
p
p
1
1
q
q
F ( , . . . , ; , . . . , ; z) , (1.13)
n=0
n
n
n=0
n
n
where z C, p q, i, j C, j 6= 0, 1, 2, . . . , (i = 1, 2, . . . , p, j = 1, 2, . . . , q).
2 INTEGRALS INVOLVING GENERALIZED GAUSS HYPERGEO-METRIC AND CONFLUENT
HYPERGEOMETRIC FUNCTION
p
p
In this section we alculate the F (,)(a, b; c; z)-function with some algebraic function.
Theorem 1 For the generalized Gauss hypergeometric function, we have the following integral
1
1
r
r
1
p
p
x (1 x) F (,) (a, b; c; kx) dx
0
p
p
= (1 , ) ·
(a)n (1 )n
(,) (b + n, c b) kn
, (2.1)
n=0
(1 )n
(b, c b) n!
where, (p) 0, and | arg (1 kx) | < < p; (c) > (b) > 0
Proof Making use of relation (1.9), it gives
n
n
1 (,) n
1
1
I = r x (1 x) · (a)
p (b + n, c b) (kx) dx
1
0
n
n
(b, c b)
(b, c b)
(,)
n=0
(b, c b) n!
1
1
n 1
= · (a)
n=0
n=0
p (b + n, c b) (k) r
n!
n!
xn (1 x) dx
n=0
0
n=0
0
p
p
·
·
(,) (b + n, c b) (k)n
= (a)n
n=0
(b, c b)
(n + 1, )
n!
p
p
·
·
(,) (b + n, c b) (k)n
(n + 1) ( )
= (a)n
p n
p n
n=0
(b, c b) n!
(n + 1)
(1 )
(1 )
= (1 ) ( )
n
n
· (a)
n=0
n=0
(,) (b + n, c b) (k)n (1 )
n!
n
n!
n
n=0
n
n=0
n
(b, c b)
(b, c b)
= (1 , ) ·
(a)n (1 )n
(,) (b + n, c b) kn
(1 )
(1 )
p
p
.
n=0
(1 )n
(b, c b) n!
This complete the proof of the Theorem 1.
If we set p = 0 in above result then we obtain the special case of (2.1) in terms of classical Gauss hypergeometric function as given in the following result:
Corollary 1.1
1
1
r
r
1
x (1 x)
0
2F1
(a, b; c; kx) dx = (1 , )
3F2
a, b, 1 ; k .
( \
( \
1
(2.2)
The integral of Gauss Confluent hypergeometric function is given by
Corollary 1.2
x (1 x) (,) (b; c; kx) dx
x (1 x) (,) (b; c; kx) dx
r
r
1
1
p
0
p
p
·
·
(,) (b + n, c b) (k)n
= (1 + n , )
n=0
(b, c b)
. (2.3)
n!
Moreover, for the generalized hypergeometric function pFq , we have the following corol- lary:
Corollary 1.3
1
1
r
r
1
x (1 x)
0
pFq
1, …, p;
1, …, q ;
kxl dx
= (1 , )
p+1
Fq+1
1, …, p, 1 ; k (2.4)
l
l
1, …, q, 1
for x C, p q; j, j C,
Theorem 2
0, 1, 2, …; i = (1, p) , j = (1, q).
1
1
r
r
p
p
x (x 1) F (,) (a, b; c; kx) dx
1
p
p
n
n
= (, ) ·
(a)n
()n
(,) (b + n, c b) kn
. (2.5)
Proof
( )
n=0
(b, c b) n!
n
n
(,) n
1
1
I = r x (x 1) · (a)
p (b + n, c b) (kx) dx
2
1
n
n
(b, c b)
(b, c b)
(,)
n=0
(b, c b) n!
1
1
n
= · (a)
n=0
n=0
p (b + n, c b) (k) r
n!
n!
x+n (x 1) dx.
n=0
1
n=0
1
Let x = 1 + t, then we arrive at
(,) n
I = · (a)
p (b + n, c b) (k) r
t1 (1 + t)+n dt,
2
using the formula
n
n=0
(b, c b)
n! 0
then we have
() () = ( + )
r
r
0
x1 (1 + x)(+) dx, (2.6)
p
p
·
·
(,) (b + n, c b) (k)n () ( n)
I2 = (a)n
n=0
(b, c b) n!
( n)
p
p
·
·
(,) (b + n, c b) (k)n
()n
= (, ) (a)n
n=0
(b, c b)
.
n! ( )n
This complete the proof of the Theorem 2.
We can also obtain special cases of Theorem 2 as done in Theorem 1.
Theorem 3
r
r
p
p
ekxx1F (,) (a, b; c; lx) dx
p
p
n
n
n
n
0
()
k
k
n
n
= · (a)
n=0
n=0
()
(,) (b + n, c b) ( l \
k
k
, k 6= 0. (2.7)
n=0
n=0
n! (b, c b)
n! (b, c b)
a
a
b1
b1
Proof Using (1.11) and (1 lxt)
= ),n=0
n! (lxt)
, we have
(a)n
(a)n
r
r
n
n
r
r
1 1
0
0
0
0
I3 = (b, c b)
tb1 (1 t)c ekxx1
1F1
; ; p
( \
( \
t(1 t)
b1
b1
× ·n=0
(a)n (lxt)n
n!
dxdt, (2.8)
·
·
1
= (b, c b) (a)
n=0
ln 1
r
r
r
r
n n! 0 0
tb+n1 (1 t)c ekxx+n1
1F1
; ; p dxdt t(1 t)
n
n
= · (a)n l
( \
( \
r ekxx+n1dx r
1
b1
b1
tb+n1 (1 t)c
F (; ; p \ dt
1 1
1 1
n=0
(b, c b) n! 0 0
·
·
r
r
n
t(1 t)
because,
=
n=0
(a)n l
(b, c b) n! 0
r
r
(
(
ekxx+n1 (,) (b + n, c b) , dx
p
p
r
r
then we have
1
e
k 0 k
+n
1 1
k
k
d = +n
0
(,)
e ()+n1 d
n
= · ( \
= · ( \
3
3
k n!
k n!
(b, c b)
(b, c b)
k
k
I (a)n()n () p (b + n, c b) l .
n=0
n=0
n=0
n=0
This complete the proof of Theorem 3.
If we set p = 0 in (2.7) then we obtain the following corollary:
Corollary 3.1
r ekxx1
0
0
F (a, b; c; lx) dx =
()
k
k
F ( a, b; ; l \ , k 6= 0. (2.9)
2 1 3 2 c k
The integral of Gauss Confluent hypergeometric function is given by the following result:
Corollary 3.2
r
r
p
p
ekxx1 (,) (b; c; lx) dx
0
p
p
(b, c b)
(b, c b)
= ·
= ·
( + n) (,) (b + n, c b) ( l \n
n=0
k n!
k
n=0
k n!
k
, k 6= 0. (2.10)
n=0
n=0
This result is in complete agreement with the result given in [6, p.98].
Next, for the generalized hypergeometric function pFq , we have the following corol- lary:
Corollary 3.3 For x C, p q; j, j C,
we have
0, 1, 2, …; (i = (1, p); j = (1, q)),
r ekxx1
F 1, …, p;
<>lxl dx = () F 1, …, p, ; l l , k 6= 0.
0
Theorem 4
r
r
p q 1, …, q ;
k p+1 q
1, …, q k
(2.11)
p
p
x1 (x + ) F (,) (a, b; c; kx) dx
0
p
p
+n
+n
= (, ) · (a) ()
( )
(,) (b + n, c b) kn
. (2.12)
Proof
n n
n=0
n
n
n
(,)
(b, c b) n!
n
I = r x1 (x + ) · (a)
p (b + n, c b) (kx) dx
4
0
n
n
(b, c b)
(b, c b)
(,)
n=0
(b, c b) n!
n
= · (a)
n=0
n=0
p (b + n, c b) (k) r
n!
n!
x+n1 (x + ) dx.
n=0
0
n=0
0
Let x = t, then we arrive at
n
n
(b, c b)
(b, c b)
n!
n!
(,) n
4
4
I = · (a)
n=0
n=0
p (b + n, c b) (k) +n r
t+n1 (1 + t) dt,
n=0
0
n=0
0
using the relation (2.6), then we have
p
p
n
n
I4 = · (a)
(,) (b + n, c b) (k)n
+n
( + n) ( n)
n=0
(b, c b) n!
·
·
()
p
p
(,) (b + n, c b) (k)n
= (, ) +n
n=0
(a)n
()n
( )n
,
(b, c b) n!
then we easily get the R.H.S. of (2.12). This complete the proof of the Theorem 4.
Theorem 5
(,)
(,)
r
r
1
(1 x) (1 + x) Fp (a, b; c; kx) dx
p
p
1
·
·
= 2++1 ( + 1, + 1)
n=0
(a)n
(,) (b + n, c b) (b, c b)
kn
F (n, + 1, + + 2; 2) n! .
(2.13)
Proof
1
(,) n
n
n
I = r (1 x) (1 + x) · (a)
p (b + n, c b) (kx) dx
5
1
n
n
(b, c b)
(b, c b)
(,)
n=0
(b, c b) n!
n
n
n 1
= · (a)
n=0
n=0
p (b + n, c b) (k) r
n!
n!
1
1
(1 x) (1 + x) x dx.
n=0
n=0
2
2
b1 a
b1 a
Now, by putting 1x = t dx = 2 dt, and using the integral representation of Gauss hypergeometric series
r
r
(c) 1
0
0
F (a, b; c; x) = (b) (c b)
tb1 (1 t)c (1 tx) dt, (2.14)
then we arrive at the following result after a little simplification:
( + 1) ( + 1)
I = 2 ·
I = 2 ·
(a)n
(,) (b + n, c b) (k)n
p
p
++1
5
( + + 2)
n=0
(b, c b) n!
× F (n, + 1, + + 2; 2) ,
then we obtain the desired result in (2.13). This completes the proof.
The next theorem considers the behavior of the generalized Gauss hypergeometric func- tion using the gamma function.
Theorem 6
lim
l
1
1
( ())
Fp(,)
(a, b; ; x) =
xl+1
(a) (b) (b l)
(a + l + r)!
× · p
× · p
(,) (b + l + r + 1, b l) xr. (2.15)
(l + r + 1)!
r=0
Proof Making use of (1.9), we have
p
p
(b, b)
(b, b)
I6 = · (a)n
n
=0
n
=0
(,) (b + n, b) xn
n!
n!
1
l ()
l ()
lim ,
n
=0
n
=0
n
n
by using (1.7), we obtain
= · r
= · r
(a)n
6
6
(b)
(b)
I
n=0
n=0
1
b1
b1
tb+n1 (1 t)
F (; ; p \ x
l(1 t
l(1 t
)
(l b)
(l b)
dt,
n=0
0
n=0
0
1
1
1
1
t (1 t)
t (1 t)
n!
n!
by setting n = l + r + 1 and using (1.7), it yields
(x)l+1
l+r+1 (,) r
l+r+1 (,) r
(a)
(b) (l b)
(b) (l b)
(l + r + 1)!
(l + r + 1)!
p
p
= · (b + l + r + 1, b l) x ,
r=0
r=0
r=0
r=0
then we arrive at the desired result in (2.14).
The special cases of (2.15) for the generalized Gauss hypergeometric function and con- fluent hypergeometric function are given in the following corollaries:
Corollary 6.1
lim
l
1
1
( ())
2F1
(a, b; ; x) = xl+1 (a)l+1 (b)l+1 F
(l + 1)! 2 1
a + l + 1, b + l + 1 ; x . (2.16)
l
l
l + 2
Corollary 6.2
1
xl+1
1 (,) (b + l + r + 1, b l) xr
p
p
lim
( ())
(,) (b; ; x) =
· .
l p
(l)
r=0
(l + r + 2)
(b, l b)
r! (2.17)
-
INTEGRALS INVOLVING GAUSS HYPERGEOMETRIC FUNC-TION WITH JACOBI POLYNOMIALS
n
n
The Jacobi polynomial P (,) (x) [13, p. 254] is defined as following:
n
n
n! 2
n! 2
P (,) (x) = (1 + )n
F n, 1 + + + n;
1 xl , (3.1)
1
1
1 + ;
1 + ;
2
2
where 2F1 is the classical hypergeometric functions; when = = 0, then the polyno- mial in (3.1) becomes the Legendre polynomial [13, p. 157].
We also have
P (,) (1) = (1 + )n .
n n!
Theorem 7 Integral formula involving Gauss hypergeometric function multiplied with
Jacobi polynomials is given by
1
µ
µ
r x (1 x) (1 + x) P (,) (x) F (,) (a, b; c; kx) dx
n p
1
n +µ+1
n +µ+1
p
p
= (1) 2 (µ + 1, n + + 1) ·
(a)r
(,) (b + r, c b) kr
n! (µ + + 1)n
r=0
(b, c b) r!
× 3F2
r, µ + + 1, µ + 1;
µ + + n + 1, µ + + n + 2;
1l . (3.2)
Proof By using (1.9), we have
1
(,) r
µ
µ
I = r x (1 x) (1 + x) P (,) (x) · (a)
p (b + r, c b) (kx) dx
7
1
(,)
n r
r=0
r 1
(b, c b) r!
= ·
= ·
µ
µ
(b, c b)
(b, c b)
n
n
(a)r p (b + r, c b) (k) r
r=0
r!
r=0
r!
x+r (1 x) (1 + x) P (,) (x) dx
1
1
r=0
1
r=0
1
Next, we use the following formula:
1
+µ+1
r (,)
r (,)
µ n 2 (µ + 1) (n + + 1) (µ + + 1)
x (1 x) (1 + x) P (x) dx = (1)
1
× 3F2
n
, µ + + 1, µ + 1;
µ + + n + 1, µ + + n + 2;
n! (µ + + n + 1) (µ + + n + 2) 1l , (3.3)
where > 1 and > 1. Also, 3F2 is the special case of generalized hypergeometric series.
n +µ+1
n +µ+1
Then we arrive at
I7 = ·
(a)r
(,) (b + r, c b) (k)r
(1) 2 (µ + 1) (n + + 1) (µ + + 1)
p
p
r=0
(b, c b) r!
n! (µ + + n + 1) (µ + + n + 2)
× 3F2
r, µ + + 1, µ + 1;
µ + + n + 1, µ + + n + 2;
1l ,
by a little simplification, then we arrive at the desired result in (3.2). This complees the proof.
-
INTEGRALS INVOLVING GAUSS HYPERGEOMETRIC FUNC-TION WITH LEGENDRE
FUNCTION
1
1
The Legendre functions are the solution of Legendres differential equation [4, sec.3.1]
2 d 2f df
(1 z ) dz2 2zdz
where z, µ, are unrestricted.
1
+ ( + 1) µ2 (1 z2)
l f = 0, (4.1)
If we substitute f = (z2 1) 2µ , then (4.1) becomes
2 d2 d
(1 z ) dz2 2 (µ + 1) z dz + [ (µ ) (µ + + 1)] = 0, (4.2)
and with = 1 1 z as the independent variable the above differential equation becomes
2 2
as following:
d2 d
(1 ) d2 + (µ + 1) (1 2) d + [ ( µ) (µ + + 1)] = 0. (4.3)
The solution of (4.1) in the form of Gauss hypergeometric type equation with a =
µ , b = µ + + 1 and c = µ + 1, as follows.
1
µ 1 ( z + 1\2µ 1 1 l
f = P (z) = (1 µ)
z 1
F , + 1; 1 µ;
2 2 z
, |1 z| < 2,
(4.4)
where Pµ (z) is known as the Legendre function of the first kind [4].
Next, we derive the integrals with Legendre function.
Theorem 8 Integral formula involving Gauss hypergeometric function multiplied with Legendre function is given as following:
1
1
r
r
µ
µ
x1 (1 x2) 2 Pµ (x) F (,) (a, b; c; kx) dx
p
µ µ
µ µ
0
p
p
= (1) 2 (1 + µ + ) ·
(a)n
(,) (b + n, c b) kn
(1 µ + )
2n ( + n)
n=0
(b, c b) n!
( + + ) (1 + + + )
( + + ) (1 + + + )
× 1 +n µ +n µ . (4.5)
Proof
2 2 2 2
2 2 2
1
(a)
(,) (b + n, c b) (kx)n
x (1 x ) P (x) ·
x (1 x ) P (x) ·
I8 = r
0
0
µ
1 2 2 µ
n=0
n p dx
(b, c b) n!
µ µ
µ µ
Next, using the formula [4, sec. 3.12] for () > 0, µ N.
1
1
r 1
µ
2 2 µ
(1) 2 () (1 + µ + )
,
,
x (1 x )
P (x) dx = (1 µ + ) (1 + + µ ) (1 +
+ µ + )
0
then we obtain
2 2 2 2
2 2 2
(4.6)
·
·
(a)
(,) (b + n, c b) (k)n
(1) 2 ( + n) (1 + µ + )
n
8
I = (b, c b)
1 +n µ +n µ .
p
p
µ µn
µ µn
n! (1 µ + ) ( + + ) (1 + + + )
n=0
This completes the proof.
2 2 2 2
2 2 2
5 INTEGRALS INVOLVING GAUSS HYPERGEOMETRIC FUNC-TION AND BESSEL
MAITLAND FUNCTION
The Bessel Maitland function (also known as Wright generalized Bessel function) de- fined as following [8]:
J (z) = (µ, + 1 : z) = ·
J (z) = (µ, + 1 : z) = ·
µ
n=0
1
(µn + + 1)
(z)
n
n
n!
. (5.1)
Theorem 9 r
x Jµ (x) F (,) (a, b; c; kx) dx
0
(a)
p
p
p
(,) (b + n, c b)
( + n + 1) kn
= · n . (5.2)
Proof
n=0
(b, c b)
(,)
(1 + µ µ ( + n)) n!
n
= ·
= ·
9
9
(b, c b)
(b, c b)
n!
n!
I (a)n p (b + n, c b) k r
n=0
n=0
x+n Jµ (x) dx,
n=0
0
n=0
0
Next, using the following formula [15]:
r x Jµ (x) dx = ( + 1)
( () > 1, 0 < µ < 1) , (5.3)
0 (1 + µ µ)
then we arrive at the desired result in (5.2). This completes the proof.
6 CONCLUDING REMARKS
We have obtained some new integrals involving Gauss hypergeometric and Confluent hypergeometric function. The results obtained here are basic in nature and are likely to find useful applications in the study of simple and multiple variable hypergeometric se-ries which in turn are useful in statistical mechanics, electrical networks and probability theory. Some important results are also given as special cases of our main results.
Acknowledgments The author (Dinesh Kumar) would like to express his deep thanks to NBHM (National Board of Higher Mathematics), India, for granting a Post-Doctoral Fellowship (sanction no.2/40(37)/2014/R&D-II/14131).
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