Theoretical Mathematics & Applications, vol.1, no.1, 2011, 115-123
ISSN: 1792- 9687 (print), 1792-9709 (online)
International Scientific Press, 2011
Finite Integrals Involving Jacobi Polynomials
and I-function
Praveen Agarwal1 , Shilpi Jain2 and Mehar Chand3
Abstract
The aim of the present paper is to evaluate new finite integral formulas involving Jacobi polynomials and I-function. These integral formulas
are unified in nature and act as key formula from which we can obtain
as their special cases. For the sake of illustration we record here some
special cases of our main formulas which are also new. The formulas establish here are basic in nature and are likely to find useful applications
in the field of science and engineering.
Mathematics Subject Classification : 33C45, 33C60
Keywords: I-function, generalized polynomials.
1
Introduction
The I-function will be defined and represented as follows [1, p. 26, Eqn.(2.1.41)]:
"
Ipm,n
i ,qi ,r
1
2
3
(aj ,αj )1,n ;(aji ,αji )n+1,p
i
z
(bj ,βj )1,m ;(bji ,βji )m+1,qi
#
1
=
2πi
Z
φ(ξ)z ξ dξ
L
Anand International College of Engineering, Jaipur-303012, India,
e-mail: goyal− praveen2000@yahoo.co.in
Poornima College of Engineering, Jaipur-303012, India
G. G. S. Group of Institutions, Talwandi Sabo, Bathinda-151302, India,
e-mail:mc75global@yahoo.in
Article Info: Revised : October 30, 2011.
Published online : October 31, 2011
(1)
116
Finite Integrals Involving Jacobi Polynomials and I-function
where
m
Y
φ(ξ) =
r
X
i=1
Γ(bj − βj ξ)
j=1
qi
"
n
Y
Γ(1 − aj + αj ξ)
j=1
#
p
Y
Y
Γ(aji − αji ξ)
Γ(1 − bji + βji ξ)
(2)
j=1
j=1
and m, n, pi , qi are integers satisfy 0 ≤ n ≤ pi , 1 ≤ m ≤ qi (i = 1...r) r is finite αj , βj , αji , βji are positive integer and aj , bj , aji , bji are complex numbers.
I-function which is a generalized form of the well known H-function [2, p.10,
Eqn.(2.1.1)] In the sequel the I-function will be studied under the following conditions of existence:
Ai > 0, |argz| <
Ai π
2
(3)
where
Ai =
n
X
j=1
αj −
pi
X
αji +
j=n+1
m
X
βj −
j=1
qi
X
βji , ∀i = (1, 2, ..., r)
(4)
j=m+1
,...,mr
The general class of polynomials Snm11,...,n
[x] introduced by Srivastava will
r
be defined and represented as follows [3, p.185, Eqn.(7)]:
[n1 /m1 ]
,...,mr
Snm11,...,n
[x] =
r
X
l1 =0
...
[nr /mr ] r
X Y
lr =0
i=1
(−ni )mi li
Ani ,li xli
li !
(5)
where n1 , ..., nr = 0, 1, 2, ...; m1 , ..., mr is an arbitrary positive integers, the
coefficients Ani ,li (ni , li ≥ 0) are arbitray constants, real or complex. On suit...mr
ably specializing the coefficients Ani ,li , Snm11...n
[x] yeilds a number of known
r
polynomials as its special cases. These includes, among other, the Bessel
Polynomials, the Lagurre Polynomials, the Hermite Polynomials, the Jacobi
Polynomials, the Gould-Hopper Polynomials, the Brafman Polynomials and
several others [4,p.158-161]
117
P. Agarwal, S. Jain and M. Chand
The following known results [5, p.945, Eqn.(16)] and [6, p.172, Eqn.(29)]
(α,β)
for the Jacobi Polynomials Pn [x] [7, p.254, Eqn.(1)], will be required in our
investigation.
Pµ(α,β) (t + y)Pµ(α,β) (t − y) =
(−1)µ (1 + α)µ (1 + β)µ
×
(µ!)2
µ
X
(−µ)n (1 + α + β + µ)n
(1 + α)n (1 + β)n
n=0
Pn(α,β) (x)tn
∞
X
1
Pn(α,β) (x)tn
(1 − t + y)−α (1 − t + y)−β = 2−α−β
y
n=0
(6)
(7)
where y denotes (1 − 2xt + t2 )1/2 in both (6) and (7).
2
Main Integrals
We establish the following integrals:
First Integral
Z
1
...mr0
(1 − x)ρ−1 (1 + x)σ−1 Snm11...n
[w(1 − x)u (1 + x)v ] ×
r0
−1
#
"
(aj ,αj )1,n ;(aji ,αji )n+1,p
i
Ipm,n
z(1 − x)h (1 + x)k dx
i ,qi ,r
(bj ,βj )1,m ;(bji ,βji )m+1,qi
[n1 /m1 ]
= 2ρ+σ−1
X
l1 =0
"
...
[nr0 /mr0 ] r0
X Y
lr0 =0
i=1
(−ni )mi li
Ani ,li wli 2(u+v)li ×
li !
#
(1−ρ−uli ,h);(1−σ−vli ,k);(aj ,αj )1,n ;(aji ,αji )n+1,p
i
(h+k) Ipm,n+2
z2
i +2,qi +1;r
(8)
(bj ,βj )1,m ;(bji ,βji )m+1,qi ;(1−ρ−σ−(u+v)li ,h+k)
The equation (8) will be converge under the conditions given in equation
(3) and
I. ρ ≥ 1, σ ≥ 1; u ≥ 0, v ≥ 0; h ≥ 0, k ≥ 0(h and k are not both zero
simultaneously)
118
Finite Integrals Involving Jacobi Polynomials and I-function
"
!#
bj
II. Re(ρ) + hmin Re
>0
βj
!#
"
bj
>0
III. Re(σ) + kmin Re
βj
Proof: To establish the integral (8), we express the I-function occurring in
its left-hand side in terms of Mellin-Barnes contour integral given by equation
(1), the integral class of polynomial occurring therein the series form given by
equation (5) and the interchange the order of summations and integration and
the order of x-and ξ-integrals (which is permissible under the conditions stated
with equation (8) and evaluating the integral with the help of a modified form
of the formula [8, p. 314, en.(3)],we easily arrive at the first integral after a
little simplification.
Second Integral:
Z 1
(1 − x)ρ−1 (1 + x)σ−1 Pµα,β (t + y)Pµα,β (t − y)×
−1
"
Ipm,n
i ,qi ,r
#
(aj ,αj )1,n ;(aji ,αji )n+1,p
i
z(1 − x)h (1 + x)k dx
(bj ,βj )1,m ;(bji ,βji )m+1,qi
µ
(−1) Γ(1 + α + µ)Γ(1 + β + µ)
×
(µ!)2
!
µ
n0
X
X
n0 + α (α + β + n0 + 1)l
(−µ)n0 (1 + α + β + µ)n0 (−n0 )l
×
0 )Γ(1 + β + n0 )
0
Γ(1
+
α
+
n
l!
(α
+
1)
n
l
0
n =0 l=0
= 2ρ+σ−1
"
Ipm,n+2
i +2,qi +1;r
(1−ρ−l,h);(1−σ,k);(aj ,αj )1,n ;(aji ,αji )n+1,p
i
(h+k) z2
#
(9)
(bj ,βj )1,m ;(bji ,βji )m+1,qi ;(1−ρ−σ−l,h+k)
where y = (1 − 2xt + t2 )1/2 , the conditions of the above result can be easily
obtained from those of first integral.
Third Integral
Z 1
1
(1 − x)ρ−1 (1 + x)σ−1 (1 − t + y)−α (1 + t + y)−β ×
y
−1
119
P. Agarwal, S. Jain and M. Chand
"
Ipm,n
i ,qi ,r
#
(aj ,αj )1,n ;(aji ,αji )n+1,p
i
z(1 − x)h (1 + x)k dx
(bj ,βj )1,m ;(bji ,βji )m+1,qi
0
= 2−α−β+ρ+σ−1
µ
n
X
X
(−n0 )l
n0 =0 l=0
"
Ipm,n+2
i +2,qi +1;r
n0 + α
n0
l!
!
(α + β + n0 + 1)l
×
(α + 1)l
(1−ρ−l,h);(1−σ,k);(aj ,αj )1,n ;(aji ,αji )n+1,p
i
#
(h+k) z2
(10)
(bj ,βj )1,m ;(bji ,βji )m+1,qi ;(1−ρ−σ−l,h+k)
where y = (1 − 2xt + t2 )1/2 , the conditions of the above result can be easily
obtained from the first integral.
To establish equation (9) and (10) the following result is required, which
the is special case of first integral:
Z
"
1
−1
m,n
(1−x)ρ−1 (1+x)σ−1 Pnα,β
0 (x)Ip ,q ,r
i i
n0
= 2ρ+σ−1
n0 + α
n0
X (−n0 )l
l=0
l!
"
Ipm,n+2
i +2,qi +1;r
!
#
(aj ,αj )1,n ;(aji ,αji )n+1,p
i
z(1−x)h (1+x)k dx
(bj ,βj )1,m ;(bji ,βji )m+1,qi
(α + β + n0 + 1)l
×
(α + 1)l
(1−ρ−l,h);(1−σ,k);(aj ,αj )1,n ;(aji ,αji )n+1,p
i
#
(h+k) z2
(11)
(bj ,βj )1,m ;(bji ,βji )m+1,qi ;(1−ρ−σ−l,h+k)
The conditions of the above result can be easily obtained from those of first
integral.
1
0
0
If we put m1 = ... = mr0 = 1;
!n1 = ... = nr = n ; w = 2 ; u = 1, v = 0 and
n0 + α (α + β + n0 + 1)l
A(n1 , l1 , ..., nr0 , lr0 ) =
in equation (8) then the
(α + 1)l
n0
polynomial Sn10 1−x
occuring therein breaks up into the Jacobi Polynomials
2
(α,β)
Pn0 [x] [9, p.68, eq.(4.3.2)] and the equation (8) reduces to the equation (11)
after a little simplification.
(α,β)
(α,β)
Proof of second integral: Put the value of Pµ (t + y)Pµ (t − y) from
equation(6) to the left hand side of equation (9) and interchanging the order
of integration and summation, then using the equation (11), we easily arrive
120
Finite Integrals Involving Jacobi Polynomials and I-function
at the required second integral after little simplification.
1
(1 − t + y)−α (1 + t + y)−β
y
from equation (7) to the left hand side of equation (10) and interchanging the
order of integration and summation, then using the equation (11), we easily
arrive at the required third integral after little simplification.
Proof of third integral: Put the value of
3
Special Cases of Main Integrals
(a) If we put r = 1, αj = βj =" αji = βji = 1 then I-function
reduces
to the gen- #
#
"
(aj ,αj )1,n ;(aji ,αji )n+1,p
(aj ,1)1,n ;(aj ,1)n+1,p
i
eral type of G-function Ipm,n
z
= Gm,n
z
p,q
i ,qi ,r
(bj ,βj )1,m ;(bji ,βji )m+1,qi
(bj ,1)1,m ;(bj ,1)m+1,q
, the equation (9) and (10) takes place in the following form:
Z 1
(1 − x)ρ−1 (1 + x)σ−1 Pµα,β (t + y)Pµα,β (t − y)×
−1
"
Gm,n
p,q
#
(aj ,1)1,n ;(aj ,1)n+1,p
z(1 − x)h (1 + x)k dx
(bj ,1)1,m ;(bj ,1)m+1,q
=
µ
n0
X
X
(−µ)n0 (1 + α + β + µ)n0
ρ+σ−1 (−1) Γ(1 + α + µ)Γ(1 + β + µ)
×
2
2
0 )Γ(1 + β + n0 )
(µ!)
Γ(1
+
α
+
n
0
! n =0 l=0
0
0
n + α (α + β + n0 + 1)l
(−n )l
×
l!
(α + 1)l
n0
µ
"
Gm,n+2
p+2,q+1
Z
(1−ρ−l,h);(1−σ,k);(aj ,1)1,n ;(aj ,1)n+1,p
(h+k) z2
#
(bj ,1)1,m ;(bj ,1)m+1,q ;(1−ρ−σ−l,h+k)
1
1
(1 − x)ρ−1 (1 + x)σ−1 (1 − t + y)−α (1 + t + y)−β ×
y
−1
#
"
(aj ,1)1,n ;(aj ,1)n+1,p
Gm,n
z(1 − x)h (1 + x)k dx
p,q
(bj ,1)1,m ;(bj ,1)m+1,q
0
= 2−α−β+ρ+σ−1
µ
n
X
X
(−n0 )l
n0 =0 l=0
l!
n0 + α
n0
!
(α + β + n0 + 1)l
×
(α + 1)l
(12)
121
P. Agarwal, S. Jain and M. Chand
"
Gm,n+2
p+2,q+1
(1−ρ−l,h);(1−σ,k);(aj ,1)1,n ;(aj ,1)n+1,p
z2(h+k) #
(13)
(bj ,1)1,m ;(bj ,1)m+1,q ;(1−ρ−σ−l,h+k)
The conditions of convergence of the above equation (12) and (13) can be obtained from those of the first integral.
(b) If we put r = 1, m = 1, n = pi = p, qi = q + 1, b1 = 0, β1 = 1, aj =
1 − aj , bji = 1 − bj , βji = βj then I-function reduces to Wright’s generalized Hy
(aj , αj )1,p
(1−aj ,αj )1,p
1,p
; −z
pergeometric function, i.e. Ip,q+1;1 z |(0,1),(1−bj ,βj )1,q = p ψq
(bj , βj )1,q
then the equation (9) and (10) reduces to the following form:
Z
1
(1 − x)ρ−1 (1 + x)σ−1 Pµα,β (t + y)Pµα,β (t − y)×
−1
p ψq
(aj , αj )1,p
; −z(1 − x)h (1 + x)k dx
(bj , βj )1,q
=
µ
n0
X
X
(−µ)n0 (1 + α + β + µ)n0
ρ+σ−1 (−1) Γ(1 + α + µ)Γ(1 + β + µ)
2
×
2
0 )Γ(1 + β + n0 )
(µ!)
Γ(1
+
α
+
n
0
! n =0 l=0
0
0
n + α (α + β + n0 + 1)l
(−n )l
×
l!
(α + 1)l
n0
µ
p+2 ψq+1
Z
(1 − ρ − l, h); (1 − σ, k); (aj , αj )1,p
; −z2(h+k)
(bj , βj )1,q ; (1 − ρ − σ − l, h + k)
(14)
1
1
(1 − x)ρ−1 (1 + x)σ−1 (1 − t + y)−α (1 + t + y)−β ×
y
−1
(aj , αj )1,p
h
k
; −z(1 − x) (1 + x) dx
p ψq
(bj , βj )1,q
!
µ
n0
0
0
X
X
n
+
α
(−n
)
(α + β + n0 + 1)l
l
= 2−α−β+ρ+σ−1
×
0
l!
(α
+
1)
n
l
0
l=0
n =0
p+2 ψq+1
(1 − ρ − l, h); (1 − σ, k); (aj , αj )1,p
; −z2(h+k)
(bj , βj )1,q ; (1 − ρ − σ − l, h + k)
(15)
122
Finite Integrals Involving Jacobi Polynomials and I-function
The conditions of convergence of the above equation (14) and (15) can be obtained from those of the first.
(c) If we put r = 1, m = 1, n = p1 = p, qi = q, b1 = 0, β1 = 1, aj = 1 − aj , bji =
bj , αj = βj = αji = βji = 1, I-function reduces to the generalized hypergeop
Y
Γ(aj )
i
h
h
i
(1−aj )1,p
j=1
a1 ,...,ap ;
1,p
=
metric function,i.e Ip,q+1
z|(0,1);(1−b
F
−
z
, then the
p q b1 ,...,bq ;
q
j )1,q
Y
Γ(bj )
j=1
equation (9) and (10) takes the following form:
Z
1
(1−x)ρ−1 (1+x)σ−1 Pµα,β (t+y)Pµα,β (t−y)p Fq
h
−1
a1 ,...,ap ;
b1 ,...,bq ;
i
− z(1 − x)h (1 + x) dx
=
µ
n0
X
X
(−1)
Γ(1
+
α
+
µ)Γ(1
+
β
+
µ)
(−µ)n0 (1 + α + β + µ)n0
2ρ+σ−1
×
2
(µ!)
Γ(1 + α + n0 )Γ(1 + β + n0 )
n0 =0 l=0
!
n0 + α (α + β + n0 + 1)l
(−n0 )l
×
l!
(α + 1)l
n0
µ
h
i
Γ(ρ + l)Γ(σ)
(1−ρ−l,h);(1−σ,k);a1 ,...,ap ;
h+k
−
z2
F
p q b1 ,...,bq ;(1−ρ−σ−l,h+k);
Γ(ρ + σ + l)
Z 1
1
(1 − x)ρ−1 (1 + x)σ−1 (1 − t + y)−α (1 + t + y)−β ×
y
−1
h
i
a1 ,...,ap ;
h
p Fq b1 ,...,bq ; − z(1 − x) (1 + x) dx
0
= 2−α−β+ρ+σ−1
µ
n
X
X
(−n0 )l
n0 =0 l=0
l!
n0 + α
n0
!
(16)
(α + β + n0 + 1)l
×
(α + 1)l
h
i
Γ(ρ + l)Γ(σ)
(1−ρ−l,h);(1−σ,k);a1 ,...,ap ;
h+k
− z2
p Fq b1 ,...,bq ;(1−ρ−σ−l,h+k);
Γ(ρ + σ + l)
(17)
The conditions of convergence of the above equation (16) and (17) can be obtained from those of the first integral.
Acknowledgements: The authors are wish to express their thanks to the
worthy referees for their valuable suggestions and encouragement.
P. Agarwal, S. Jain and M. Chand
123
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