Gen. Math. Notes, Vol. 25, No. 2, December 2014, pp. 110-124
ISSN 2219-7184; Copyright © ICSRS Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
Convolution Integral Equation of Fredholm
Type with Certain Binomial and
Transcendental Functions
V.B.L. Chaurasia1 and R.C. Meghwal2
1
Department of Mathematics, University of Rajasthan
Jaipur-302055, Rajasthan, India
E-mail: drvblc@yahoo.com
2
Department of Mathematics, Government P.G. College
Neemuch-458441, M.P., India
E-mail: meghwal66@gmail.com
(Received: 7-8-14 / Accepted: 16-9-14)
Abstract
We derive a solution of a certain class of convolution integral equation of
Fredholm type whose kernel involve certain product of binomial and special
functions by using Riemann-Liouville and Weyl fractional integral operators.
Certain interesting special cases have also been discussed.
Keywords: Riemann Liouville fractional integral, Weyl fractional integral,
Mellin transform technique, H -function.
1
Introduction
The following is the special case of Raizada’s [13] generalized polynomial set and
defined as:
Convolution Integral Equation of Fredholm…
α,β, 0
Sn
n
[ x ; r, q, A, B, k, ℓ] = (Ax + B) −α exp (β x r ) Tk,ℓ [( ax + b) α +qn exp(−βx r )]
∑
=
111
e,p,u,v
φ1 ( e, p, u, v) x L ,
… (1.1)
Where
(−1)p (−v)u (−p)e (α)p (−α − qn)  α + k + rv 
p v
e
φ1(e, p, u, v) = (Bqn−p ) ℓ n

 Aβ ,
u ! v! e ! p !
(1 − α − p)e 
ℓ
n
… (1.2)
L = ℓn + p + rv , (p, v = 0,1,..., n)
… (1.3)
And
∑
e,p,u,v
=
n
v
n
p
v =0
u =0
p =0
e=0
∑ ∑ ∑ ∑
.
… (1.4)
The polynomial set defined by (1.1) is a very general in nature and it unifies and
extends a number of classical polynomials introduced and studied by various
research workers such as Chatterjea [1, 2], Gould and Hopper [7], Krall and Frink
[11], Srivastav and Singhal [15] etc. We have made an effort, in the present paper,
to obtain an exact solution of the following convolution integral equation of
Fredholm type
∞
∫0
x
y −µ u   f(y) dy = g(x) , (x > 0)
 y
…(1.5)
where g is a recommended function, f is a unknown function to be determined and
the kernel u(x) is given by
α ,β, 0
u(x) = (cx t + d) θ S n
×H
M, N
P,Q
 λ
ω x

M ,N 
[z x ρ ; r, q, A, B, k, ℓ] H P 1,Q 1  t(x σ )
1 1 

( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
(b j ,β j )1,M ; ( b j ;β j ;B j )M +1,Q
M ,N

,

( e P ,E P ) 
1 1
( fQ ,FQ ) 
1 1 

…(1.6)
in (1.6) the H P 1,Q 1 [z] is the well known Fox’s H-function [6] and its series
1 1
representation is given by
112
V.B.L. Chaurasia et al.
∞ M1
M1, N1  (eP1 ,E P1 ) 
H P ,Q  z ( f , F  =
1 1 
 Q1 Q1  G =0 g =1
∑ ∑
( −1) G
η
φ 2 ( ηG ) z G ,
G ! Fg
…(1.7)
where
M1
∏
j=1
j ≠g
φ 2 ( ηG ) =
Q
∏
N1
Γ( f j − FjηG ) ∏ Γ(1 − e j + E jηG )
j=1
Γ(1 − f j + FjηG )
j= M1+1
P1
∏
Γ( e j − E j η G )
j= N1+1
and
ηG =
( f g + G)
Fg
The H -function defined by Inayat-Hussain [9, 10] in (1.6) as:
H
M, N
P,Q
[z] = H
=
M, N
P,Q

z


(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q 

(a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
1 + i∞
φ 3 (s) z s ds ,
∫
2πi −i∞
…(1.8)
where
M
φ 3 (s) =
∏
j=1
Q
N
Γ( b j − β js) ∏ {Γ(1 − a j + α js)}
∏
j= M +1
Aj
j=1
{Γ(1 − b j + β js)}
Bj
P
∏
j= N +1
…(1.9)
Γ( a jα js)
which contains fractional powers of some of the Γ-functions. Here z may be real
or complex but not equal to zero and an empty product is interpreted as unity.
P,Q,M and N are integers such that 1 ≤ M ≤ Q, 0 ≤ N ≤ P, αj (j = 1,…,P), βj (j =
1,…,Q) are complex numbers. The exponents Aj (j = 1,…, N) and Bj (j =
M+1,…,Q) can take non integer values, when these exponents take integer values,
the H -function reduces to the familiar H-function due to Fox.
Srivastav [16] introduced the general class of polynomial as follows
[n1 / m1 ] ( − n )
1 m1k1
m1
S n [I] =
1
k1 !
k1 =0
∑
k
I 1 An
1,k1
, :[n1 = 0,1,2,...]
…(1.10)
Convolution Integral Equation of Fredholm…
113
where m1 is an arbitrary positive integer and the coefficients A n
1, k1
( n1 , k1 ≥ 0)
are arbitrary constants, real or complex.
Let J denote the space of all functions f which are defined on R+ = [0,∞] and
satisfy
(i)
(ii)
(iii)
f ∈C ∞ ( R + )
lim [ x k f r ( x)] = 0, for all non negative integer k and r.
x →∞
f(x) = 0(1) as x → 0
J corresponds to the space of good functions defined on the whole real line (Miller
[12]).
For the present study, we shall also require the Riemann-Liouville fractional
integral (of order µ) defined by
−µ
D −µ {f(x)} = 0 D x {f(x)}
=
1 x
( x − w) µ −1 f(w) dw, (Re(µ ) > 0 ; f ∈J),
∫
Γ( µ ) 0
…(1.11)
and the Weyl fractional integral (of order h) defined by
−h
W −h {f(x)} = x D ∞ {f(x)}
=
2
1 ∞
( ξ − x) h −1 f(ξ) dξ , ( Re(h) > 0 ; f ∈J).
∫
Γ( h) x
…(1.12)
Preliminary Results
Lemma 1: Assuming the following that
(i)
P,Q,M,N are integers such that 1 ≤ M ≤ Q, 0 ≤ N ≤ P, αj (j = 1,…,P),
βj (j = 1,…,Q) are complex numbers.
(ii)

bj 

Re(µ ) > Re (h) ; Re h + ρL + tR + σηG + λ  > 0,

βj 


where j = 1,…,M, M is a positive integer,
λ ≥ 0, ( L = ℓn + p + rv), p, v = 0,1,..., n ; w ≥ 0, σ ≥ 0;
114
V.B.L. Chaurasia et al.
| arg w | <
(iii)
1
π Ω 2 , where
2
M
N
j=1
j=1
Ω 2 = ∑ | β j | + ∑ | A jα j | −
Q
∑
j= M +1
| B jβ j | −
P
∑
j= N +1
| α j | > 0;
then

y

θ
ρ
 M ,N   x  σ
 x   α,β, 0   x 
1 1
c  + d  S n z  ; r, q, A, B, k, ℓ  H P ,Q T 
1
1   y
y
y
  

   

h −µ  −µ 
W
×H
=y
−h
M, N
P,Q
  x λ
 w  
  y 


( b j ,β j )1,M ; ( b j ,β j ; B j )M +1,Q

( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
∞
M1
∞
∑ φ1(e, p, u, v) z ∑ ∑ ∑
L
G=0 g=1 R =0
e,p,u,v
  x λ
 w  
×H
P +1,Q +1   y 

M, N +1
( e P ,E P ) 
1 1 
( fQ ,FQ )
1 1 




(−1)G
η
c
φ2 (ηG ) T G d θ  
G ! Fg
d
R
 θ  x 
  
 R  y 
ρL+σηG +tR

 . …(2.1)
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q , (1−µ −ρL −σηG − tR,λ ;1)

(1− h −ρL −σηG − tR,λ;1), ( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
Proof: Making use of definition (1.12) and the series representation (1.1) and for
the generalized polynomial set and the Fox’s (See skibinski [14]) respectively in
the left hand side of (2.1), then expressing H -function in Mellin-Barnes type
contour integral and changing the order of summations and integration (which is
justified under the condition stated), we find that left hand side of (2.1).
∞
1
=
∑ φ (e, p, u, v) ∑
Γ(µ − h)2πi e,p,u,v 1
G =0
×∫
i∞
− i∞
s
x
φ 3 (s) w  
y
ρL+ σηG + tR + λs
M1
∞
∑ ∑
g =1 R =0
( −1) G
η
c
φ 2 ( ηG ) T G z L d θ  
G ! Fg
d
R
θ
 
R
 ∞

µ − h −1 −µ −ρL−σηG −λs − tR
δ
dδds.
∫ ( δ − y)
y

…(2.2)
Now evaluating the inner δ integral in (2.2) with the help of known result Erdélyi
et al. [5] and the reinterpreting the resulting Mellin-Barnes contour integral in
terms of H -function, we arrive at the desired result (2.1).
Lemma 2: Under the conditions stated with Lemma 1, we have
Convolution Integral Equation of Fredholm…
∞
∫0
y
∞
−h
  x λ
 w  
×H
P +1,Q +1   y 

M, N +1
(−1)G
η
c
φ2 (ηG )T G dθ  
G ! Fg
d
R

 f(y) dy
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q , (1−µ −ρL −σηG − tR,λ ;1)

(1− h −ρL−σηG − tR,λ ;1), ( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
θ
0
M, N
P,Q
  x λ
 w  
  ξ 
ρL+σηG+tR
 θ  x 
  
 R  y 
t

 α,β, 0   x  ρ
 M ,N   x  σ
−µ  x 
ξ c  + d  S n z  ; r, q, A, B, k, ℓ  H P 1,Q 1  t  
  ξ 

  ξ 
 1 1   ξ 
∞
×H
∞
G=0 g=1 R=0
e,p,u,v
=∫
M1
∑ φ1(e,p,u,v) z ∑ ∑ ∑
L
115
( e P ,E P ) 
1 1 
( fQ ,FQ )
1 1 


 D (h −µ ) f(ξ ) dξ, provided f ∈ J and x > 0
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q 

( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
…(2.3)
Proof: Let ∇ denote the left hand side of (2.3). Then using Lemma 1 and applying
(1.12), we get
∇=
θ

t
∞


x
 1
( ξ − y) µ−h −1 ξ −µ c  + d 

∫
  ξ 

 Γ(µ − h) y

∞
∫0
α ,β, 0
× Su
×H
=
∞
∫0
×H
M, N
P,Q
  x ρ
 M ,N   x  σ
z  ; r, q, A, B, k, ℓ  H P 1,Q 1 T 
  ξ 
 1 1   ξ 
  x λ
 w  
  ξ 
( eP ,E P ) 
1 1 
(fQ , FQ )
1 1 

 
 dξ f(y) dy
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q 
 
( a j ,α j ;A j )1, Ν ; ( a j ,α j ) N +1,P
θ
t

 α,β, 0   x  ρ
 M ,N   x  σ
−µ  x 
ξ c  + d  S n z  ; r, q, A, B, k, ℓ  H P 1,Q 1 T 
  ξ 

  ξ 
 1 1   ξ 
M,N
P,Q
  x λ
w 
  ξ 
…(2.4)
( e P ,E P ) 
1 1 
(fQ , FQ )
1 1 

  ∞ (ξ − y)µ−h −1

 ∫
f(y) dydξ, …(2.5)
(b j,β j )1,M ; ( b j,β j ;B j )M+1,Q  y
Γ(µ − h)


( a j ,α j ;A j )1, Ν ; ( a j ,α j )N+1,P
116
V.B.L. Chaurasia et al.
where it has been assumed that the change of order of integration is permissible as
in the proof of lemma 1. Now by using (1.11), we get easily the right hand side of
(2.3).
Main Theorem
If f ∈J, D µ−h {f} exists, λ > 0, x > 0, | arg w | <
π
Ω , Ω > 0, Re(µ ) > Re(h) > 0,
2 2 2
then the solution of the integral equation (1.5) is given by
c + iγ
x −s φ 4 (s)
λ µ −1
f(x) =
x
lim ∫
ds ,
γ → ∞ C−iγ ψ( s, z, T, w)
2πi
…(2.6)
where
φ 4 (s) = ∫
∞
x s−1 g(x) dx,
…(2.7)
0
and
ψ(s, z, T, w) =
∑
e,p,u,v
φ1 (e, p, u, v) z
∞
L
M1
∞
∑∑ ∑
G =0 g =1 R =0
( −1)G
η
c
φ 2 (ηG ) T G d θ  
G ! Fg
d
 − ρL − s − σηg − tR 
s + ρL + σηG + tR 
 − 
,
φ5 


 w 
λ
λ



R
θ
 
R
…(2.8)
provided that
 bj 
 (1 − a j ) 
 s + ρL + σηG + tR 
 < min Re 
− min Re   < Re 

 1≤ j ≤ N
β 
1≤ j≤M
λ
Aj 



j


 
where L = ℓn + p + rv : p, v = 0,1,..., n.
Replace f by D µ−h {f} in left hand side of (2.3), we have
g(x) = ∫
∞
0
x
×  
y
y
−h
∑
e,p,u,v
ρL + σηG + tR
φ1 (e, p, u, v) z
∞
L
  x λ
 w  
H
P +1,Q +1   y 

µ −h
× D {f(y)} dy.
M, N +1
M1
∞
∑∑ ∑
G=0 g=1 R =0
( −1)G
η
c
φ2 (ηG ) T G d θ  
G ! Fg
d
R
θ
 
R


(b j ,β j )1,M , ( b j ,β j ;B j )M +1,Q , (1−µ −σηG −ρL − tR,λ ;1) 

(1− h −ρL −σηG − tR ,λ ;1),( a j ,α j ;A j )1, N ,( a j ,α j ) N +1,P
…(2.9)
Convolution Integral Equation of Fredholm…
117
Now taking Mellin transform of both sides, we get
φ 4 (s) = ∫
∞
0
×y
∑
e,p,u, v
φ1 ( e, p, u, v) z L
∞
M1
∞
∑ ∑ ∑
G =0 g =1 R =0
( −1) G
η
c
φ 2 ( ηG ) T G d θ  
G ! Fg
d
 ∞ (s+ρL +ση + tR)−1 M, N +1   x  λ
G
 w  
H
∫ x
P +1,Q+1   y 
0


−h −ρL −σηG − tR 
( a j ,α j ;A j )1, N , ( a j ,α j ) N +1,P
(1−µ −σηG −ρL− tR,λ ;1
  (µ−h)
{f(y)} dy ,
 dx  D
 
R
θ
 
R
(1− h −ρL −σηG − tR,λ;1),
( b j ,β j )1,M , ( b j ,β j ;B j )M +1,Q ,
…(2.10)
Now by using elementary property of H -function, assuming the change of order
of integration is permissible under the conditions stated with the theorem and on
evaluating the inner integral, (2.10) reduces to
λ φ 4 (s) Γ(µ − s)
= M  y1−h D µ −h {f(y)}: s
Γ( h − s) ψ(s, z, T, w)


…(2.11)
which on applying Mellin inversion theorem gives
D µ−h {f(y)} =
c + iγ
Γ(µ − s) φ 4 (s) ds
λ
lim ∫
y h −s−1
2πi γ → ∞ c−iγ
Γ( h − s) ψ(s, z, T, w)
…(2.12)
Now operating upon both the sides by D h −µ defined by (1.11) and then on
changing the order of integration which is permissible under the conditions stated
with the theorem, we obtain
c + iγ
λ
Γ(µ − s)
f(y) =
lim ∫
2πi Γ(µ − h) γ → ∞ c−iγ Γ( h − s) Ψ (s, z, T, w)
 y h −s−1

× ∫ ξ
( y − ξ ) µ−h −1 dξ φ 4 (s) ds
0

…(2.13)
which on evaluating the inner integral by appealing to the well known definition
of Beta function, finally yield the required result (2.6).
118
V.B.L. Chaurasia et al.
Lemma 3: If
P, Q, M, N are integers such that 1 ≤ M ≤ Q, 0 ≤ N ≤ P, α j ( j = 1,..., P),
(i)
β j ( j = 1,..., Q) are complex numbers;
Re(µ ) > Re(h) ; Re(h + ρL + σηG + tR + δk 1 + λ
(ii)
bj
Bj
) > 0,
where j = 1,…,M, M is a positive integer,
σ ≤ 0, λ ≥ 0, L (ℓn + p + rv), (p, v = 0,1,..., n);
m1 be an arbitrary positive integer and the coefficients A n
(iii)
1, k1
( n1 , k1 ≥ 0 )
be arbitrary constants, real or complex;
1
| arg w | < Ω 2 where Ω 2 is defined by
2
(iv)
M
N
j=1
j=1
Q
Ω 2 = ∑ | β j | + ∑ | A jα j | −
∑
j= M +1
| B jβ j | −
P
∑
j= N +1
|α j | > 0
Then
W

y

h −µ  −µ
M1, N1
H P ,Q
1 1
=y
−h
α,β, 0
Sn
  x σ
T 
  y 

∑
e,p,u, v
  x ρ
   x t
 m   x δ 
z  ; r, q, A, B, k, ℓ  c  + d  S n 1  A  
  y 
   y 
 1   y  
( eP , E P ) 
1 1
( fQ ,FQ
1 1
φ1 ( e, p, u, v) z
λ
M, N   x 
H
w 
 P,Q   y 


L
[ n1 / m1 ] ( − n
∑
  x λ
 w  
H
P +1,Q +1   y 

M, N +1
1 m1k1
k1 !
k1 =0
( −1) G
η
c
×
φ 2 ( ηG ) T G d θ  
G ! Fg
d
)
R
 θ  x 
   
 R  y 






(b j ,β j )1,M , ( b j ,β j ;B j )M +1,Q 

( a j ,α j ;A j )1, N ,( a j ,α j ) N +1,P
k
I 1A
β z
v
n1,k1
rv − n
∞
M1
∞
∑ ∑ ∑
G =0 g =1 R =0
ρL + σηG + tR +δk1


( b j ,β j )1,M ,( b j ,β j ;B j )M +1,Q , (1−µ −ρL −σηG − tR −δk1,λ ;1) 

(1− h −ρL −σηG − tR −δk1,λ ;1), (a j ,α j ;A j )1, N , ( a j ,α j ) N +1,P
…(2.14)
Convolution Integral Equation of Fredholm…
119
Proof: To prove Lemma 3, firstly we use the definition of Weyl fractional integral
given in (1.12), expres the generalized polynomial set, H-function in series
representation, a general class of polynomials and H -function, then we change the
order of summations and integration (which is justified under the stated
conditions), evaluate the integral and reinterpreting the resulting Mellin-Barnes
contour integral in terms of H -function, we get the required result.
3
Special Cases
(1) If we set n = q = k = B = 0, ℓ = r = − 1 and A = 1 in (2.1), (2.3), (2.6) and
(2.14), we get the following results
(1.a)
θ

  x t

  x  σ ( e P ,E P ) 
M1, N1

h −µ
−µ
W  y c  + d  H P ,Q T  ( f 1 ,F 1 
1 1   y
Q1 Q1 
  y 





λ
M, N   x  ( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P 
 w  

×H
P,Q   y  (b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q 


=y
−h
∞
M1
∞
∑ ∑ ∑
G =0 g =1 R =0
( −1) G
η
c
φ 2 ( ηG ) T G d θ  
G ! Fg
d
  x λ
 w  
×H
P +1,Q+1   y 

M, N +1
R
 θ  x 
   
 R  y 
σηG + tR


(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q , (1−µ −σηG − tR,λ;1) 

(1− h −σηG − tR,λ ;1), ( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
…(3.1)
(1.b)
∞
∫0
y
−h
∞
M1
∞
∑ ∑ ∑
G =0 g =1 R =0
( −1) G
η
c
φ 2 ( ηG ) T G d θ  
G ! Fg
d
  x λ
 w  
⋅H
P +1,Q +1   y 

M, N +1
=
∞
∫0
θ
R
 θ  x 
   
 R  y 
σηG + tR

 f(y) dy
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q , (1−µ −σηG − tR,λ;1) 

(1−h −σηG − tR,λ;1), ( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
σ
  x t

M1, N1   x 
−µ
ξ c   + d  H P ,Q T  
1 1  ξ
  ξ 




( f Q ,FQ ) 
1 1 
( e P ,E P )
1 1
120
V.B.L. Chaurasia et al.
×H
M, N
P,Q
  x λ
 w  
  ξ 

 D h −µ f(ξ ) dξ.
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q 

( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
…(3.2)
(1.c)
∞
∫0
θ
t


  x σ
M1, N1
−µ  x 
y c  + d  H P ,Q T 
1 1   y
  y 


×H
M, N
P,Q
  x λ
 w  
  y 
( e P ,E P ) 
1 1 
( f Q ,FQ
1 1 


 f(y) dy = g(x)
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q 

( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
has its solution given by
 ∞
c + iγ
λx µ−1
−s
f(x) =
Lt
x ∑
2πi γ → ∞ ∫c−iγ
G =0
 − s − σηG − tR 
w
× φ 5 

λ


M1
∞
∑ ∑
g =1 R =0
( −1) G
η
c
φ 2 ( ηG ) T G d θ  
G ! Fg
d
 −s −σηG − tR  




λ


R
θ
 
R
−1
 φ 4 (s) ds.

…(3.3)
(1.d)
θ

t
δ
σ


m1   x   M , N   x 
(h −µ )  −µ  x 
W
 y c  + d  S n  A   H P 1,Q 1 T 
1  y 
1 1   y
  y 






×H
=y
−h
M, N
P,Q
[ n1 / m1 ] ( − n
∑
k1 =0
  x λ
 w  
  y 
)
1 m1k1
k1 !
k
I 1A
φ 2 ( ηG
∞
M1
∞
∑ ∑ ∑
n1,k1
G =0 g =1 R =0
η

) T G dθ
  x λ
 w  
×H
P +1,Q+1   y 

M, N +1


(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q 

( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
c
 
d
R
( e P ,E P ) 
1 1 
( fQ ,FQ
1 1 




( −1) G
G ! Fg
 θ  x 
   
 R  y 
σηG + tR +δk1

 …(3.4)
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q , (1−µ −σηG − tR −δk1 ,λ;1) 

(1− h −σηG − tR −δk1,λ ;1), ( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
Convolution Integral Equation of Fredholm…
121
(2) If we set A = 1, B = q = k = 0 and ℓ = −1 in (2.1), (2.3), (2.6) and (2.14), we
obtain the following
(2.a)

W
h −µ  −µ
y


×H
=y
−h
n
θ
ρ
  x t

 M ,N   x  σ
(r)   x 
c  + d  H n z  ; α, β H P 1,Q 1 T 
  y 

  y 
 1 1   y 
M, N
P,Q
n
∑ ∑
  x λ
 w  
  y 
( − v) u ( −α − ru) n
u !v!
v =0 u =0
φ 2 ( ηG
  x λ
 w  
×H
P +1,Q+1   y 

M, N +1


(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q 

( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
β z
v
rv− n
∞
M1



∞
∑ ∑ ∑
G =0 g =1 R =0
η

) T G dθ
c
 
d
R
 θ  x 
   
 R  y 
( e P ,E P ) 
1 1 
( fQ ,FQ )
1 1 

( −1) G
G ! Fg
σηG + tR +ρ( rv − n)

 …(3.5)
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q , (1−µ −σηG − tR −ρrv +ρn,λ;1) 

(1− h −σηG − tR -prv+ρn,λ;1), ( a j ,α j )1, N ; ( a j ,α j ;A j ) N +1,P
(2b)
∞
∫0
θ
t
ρ


 M ,N   x σ
(r)   x 
−µ  x 
ξ c  + d  H n  z  ; α, β H P 1,Q 1 T 
  y 

  ξ 
 1 1   ξ 
×H
n
= y −h ∑
M, N
P,Q
n
∑
v =0 u =0
  x λ
 w  
  ξ 

 D h −µ f(ξ ) dξ
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q 

( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
( − v) u ( −α − ru) n
u !v!
φ 2 ( ηG
( eP ,E P ) 
1 1 
( fQ , FQ
1 1 

β v z rv−n
∞
M1
∞
∑ ∑ ∑
G =0 g =1 R =0
η

) T G dθ
c
 
d
R
 θ  x 
   
 R  y 
( −1) G
G ! Fg
σηG + tR +ρ( rv − n)
122
V.B.L. Chaurasia et al.
  x λ
 w  
×H
P +1,Q+1   y 

M, N +1

 f(y) dy
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q , (1−µ −σηG − tR −ρrv +ρn,λ ;1) 

(1− h −σηG − tR −ρrv+ρn,λ;1), ( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
…(3.6)
(2.c)
∞
∫0
θ
ρ
  x t

 M , N   x  σ ( eP ,E P ) 
(r)   x 
y c  + d  H n z  ; α, β H P 1,Q 1 T  ( f 1 ,F 1 ) 
  y 

  ξ 
 1 1   y  Q1 Q1 
λ
M, N   x  ( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P 
 w  
 f(y) dy = g(x)
×H
P,Q   y  (b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q 


−µ
has its solution given by
c + iγ
 n
λ x µ−1
−s
f(x) =
lim
x ∑
2πi γ → ∞ ∫c−iγ
 v=0
∞
M1
∞
∑ ∑ ∑
G =0 g =1 R =0
v
( − v) u ( −α − ru) n
u =0
u !v!
∑
( −1) G
η
c
φ 2 ( ηG ) T G d θ  
G ! Fg
d
 − s − σηG − tR − ρrv + ρn 
w
× φ 5 

λ


R
β v z rv−n
θ
 
R
 -s −σηG − tR −ρrv+ρn  




λ


−1
 φ 4 (s) ds.

…(3.7)
(2.d)
θ
 
t
 (r)   x ρ
 m1   x δ  M ,N   x σ


x

h−µ
−µ
W y c  + d Hn z  ; α, β Sn A   HP 1,Q 1 T 

  y 
 1   y   1 1   y 
   y 

  x  λ ( a j ,α j;A j )1, N ;( a j,α j ) N +1,P  
 w  

×H
P,Q   y  (b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q 

 
n
n [ n1 / m1] ( −v) ( −α − ru) ( −n )
∞
u
n
1 m1k1 k
−h
v rv−n
1
=y ∑ ∑ ∑
I A n ,k β z
∑
1 1
u ! v!
k1 !
v=0 u =0 k =0
G =0
( eP ,EP ) 
1 1 
( fQ ,FQ )
1 1 

M, N
1
( −1) G
η
c
φ2 (ηG ) T G d θ  
G ! Fg
d
R
 θ  x 
   
R y 
M1
∞
∑ ∑
g =1 R =0
σηG + tR+δk1+ρ( rv−n)
Convolution Integral Equation of Fredholm…
  x λ
 w  
×H
P +1,Q+1   y 

M, N +1
123

. …(3.8)
(b j ,β j )1,M ; ( b j ,β j ;B j )M +1,Q , (1−µ −σηG − tR −δk1 −ρrv +ρn,λ ;1) 

(1− h −σηG − tR −δk1−ρrv +ρn,λ;1), ( a j ,α j ;A j )1, N ; ( a j ,α j ) N +1,P
(3) The result obtained by Srivastava, H.M. and Raina, R.K. [17] follows as
special cases of our result on assigning certain values to parameters in the
function involved and by setting θ = 0.
(4) On taking A j ( j = 1,..., N) = B j ( j = M + 1,..., Q) = 1, σ → 0 and θ = 0 in (2.1)
and (2.3) the result reduce to known result derived by Goyal, S.P. and Mukherjee,
Rohit [8] with t = 0.
(5) Letting A j ( j = 1,..., N) = B j ( j = M + 1,..., Q) = 1, σ → 0, n = q = k = B = 0,
ℓ = r = − 1, A = 1 and θ = 0, the result reduce to a known result derived by
Chaurasia, V.B.L. and Patni, Rinku [3] ]with n = 0.
(6) Letting A j ( j = 1,..., N) = B j ( j = M + 1,..., Q) = 1, σ → 0, n = q = k = B = 0,
ℓ = r = − 1, A = 1 and θ = 0, the result reduces to a known result derived by
Chaurasia, V.B.L. and Shekhawat, Ashok Singh [4] with n = 0.
(7) Taking A j ( j = 1,..., N) = B j ( j = M + 1,..., Q) = 1, σ → 0 and θ = 0 in (2.6),
we find a known result of Goyal, S.P. and Mukherjee, Rohit [8] with t = 0.
Acknowledgement
The authors are grateful to Professor H.M. Srivastava, University of Victoria,
Canada for his kind help and valuable suggestions in the preparation of this paper.
References
[1]
[2]
[3]
[4]
S.K. Chatterjea, Some operational formulas connected with a function
defined by a generalized Roderigues formula, Acta Math. Acad. Hunger,
17(1966), 379.
S.K. Chatterjea, Quelques functions génératrics des polynomes d’Hermite,
du point del’vue de algebre de Lie, C.R. Acad. Sci. Paris, Ser. A-B
268(1969), A 600.
V.B.L. Chaurasia and R. Patni, Integral equation of Fredholm type with
special functions, Indian J. Pure Appl. Math., 32(2001), 747.
V.B.L. Chaurasia and A.S. Shekhawat, Fredholm type integral equations
and certain polynomials, Kyungpook Math. J., 45(2005), 471-481.
124
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
V.B.L. Chaurasia et al.
A. Erdélyi et al., Tables of Integral Transforms, McGraw-Hill, New York,
(1954), 201.
C. Fox, The G and H-functions as symmetrical Fourier kernels, Trans.
Amer. Math. Soc., 98(1961), 395-429.
H.W. Gould and A.T. Hopper, Operational formulas connected with two
generalizations of Hermite polynomials, Duke Math. J., 29(1962), 51.
S.P. Goyal and R. Mukherjee, A class of convolution integral equations
involving product of generalized polynomial set, general class of
polynomials and Fox’s H-function, Bull. Cal. Math. Soc., 96(2004), 265272.
A.A. Inayat-Hussain, New properties of hyper geometrical series derivable
from Feynman integrals-I: Transformation and reductions formula, J.
Phys. A. Math. Gen., 20(1987), 4109-4117.
A.A. Inayat-Hussain, New properties of hyper geometric series derivable
from Feynman integrals-II: A generalization of the H-function, J. Phys. A.
Math. Gen., 20(1987), 4119-4128.
H.L. Krall and O. Frink, A new class of orthogonal polynomials: The
bessel polynomials, Trans. Amer. Math. Soc., 65(1949), 100.
K.S. Miller, The Weyl Fractional Calculus in e Fractional Calculus and
its Applications, B. Ross (ed.), Springer Verlag, Berlin, 80(1975).
S.K. Raizada, Ph.D. Thesis, Bundelkhand University, (1991).
P. Skibi n ski, Some expansion theorems for the H-function, Ann. Polon.
Math., 23(1970), 125.
H.M. Srivastava and J.P. Singhal, A class of polynomials defined by
generalized Rodrigues formula, Ann. Mat. Pure Appl., 90(1971), 75.
H.M. Srivastav, A contour integral involving Fox’s H-functions, Indian J.
Math., 14(1972), 1-6.
H.M. Srivastav and R.K. Raina, On certain methods of solving a class of
integral equations of Fredholm type, The Journal of the Australian
Mathematical Society, Ser. A 52(1992), 1-10.