Gen. Math. Notes, Vol. 16, No. 1, May, 2013, pp. 40-48
ISSN 2219-7184; Copyright © ICSRS Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
The Inversion of an Integral Equation
Pertaining to a Generalized Polynomial Set
V.B.L. Chaurasia1 and J.C. Arya2
1
Department of Mathematics, University of Rajasthan
Jaipur-302004, India
E-mail: drvblc@yahoo.com
2
Department of Mathematics, Govt. Post Graduate College
Neemuch-458441 (M.P.), India
E-mail: profarya76@gmail.com
(Received: 4-2-13 / Accepted: 5-3-13)
Abstract
The main object of this paper is to evaluate a certain class of convolution
integral equation of Fredholm type with n-generalized polynomial sets. Here, the
integral equation is solved by applying the Mellin transform.
Keywords: Mellin Transform, convolution integral equation, generalized
polynomial set.
1
Introduction
This paper deals with the investigation of the inversion of the integral
∏
i =1,...,n
g i ( x) =
∞
∏ ∫0
i =1,...,n
x
 dy 
h i   f i ( y)  , ( x > 0),
y
 y
…(1)
The Inversion of an Integral Equation…
41
Where gi is a prescribed function, fi is an unknown function to be determined and
the kernel hi is given by
 ( a x αi + b ) pi ( c x βi + d ) qi
i
i
i
Kn
∏ h i ( x) = ∏  i
ξi
i
−
t
x
i =1,...,n
i =1,...,n
e i

p i ,q i
. Rn
i
[ x, a i , b i , c i , d i ; α i , β i , γ i , δ i ; e
−ti x
ξi




ξ
 θi
ni
αi
pi + γ i n i
βi
q i + δi n i − t i x i 
= ∏ {x ( φ + xD x )} {( a i x + b i )
(ci x + d i )
e
}
i =1,...,n 

…(2)
q
[ x] is introduced by Agrawal and Chaubey [1] by
Where the polynomial set R p,
n
means of the following formula
q
q
R p,
[ x] = R p,
[ x, a, b, c, d ; α, β, γ, δ ; ω( x)]
n
n
=
( ax α + b) − p ( cx β + d) −q n
Tθ,φ [( ax α + b) p + γn ( cx β + d) q +δn ω( x)], n = 0,1,2,...
K n ω( x)
…(3)
Where
Tθn,φ = x θ ( φ + x D x ) , D x =
d
dx
…(4)
{K n }∞n =0 is a sequence of constants, a,b,c,d, α,β,γ,δ,p,q are constants and ω(x) is
any general function of x, differentiable an arbitrary number of times.
The polynomial R np,q [ x] is general in nature and yields a number of known
polynomials as its special cases. In particular α = β = 1, Kn = n!, φ = 0, θ = &1,
the polynomial set R np,q [ x] reduces to S np,q [ x, a, b, c, d; γ, δ; ω( x)], given by
Srivastava and Panda [6].
42
V.B.L. Chaurasia et al.
2
Lemma
∏
We begin with Lemma involving the Mellin transform of
i =1,...,n
h i ( x) which is as
follows:
If
∏
i =1,...,n
H i (s) =
∏
i =1,...,n
M {h i ( x) ; s } where
 ni
∏ H i (s) = ∏  ∑
i =1,...,n
i =1,...,n k i =0

.
( −q i − δ i n i ) m
mi !
i
(φ)
n i −ki
pi + γ i n i qi + δi n i
∑ ∑
ℓ i =0
(bi )
( −1)
∏
i =1,...,n
h i ( x) is defined by (2), then
k i + ℓ i + mi
( −n i ) k ( − p i − γ i n i ) ℓ
i
ki !
mi =0
pi + γ i n i − ℓ i
(d i )
q i + δi n i − m i
i
ℓi !
m
ℓ
(a i ) i ( c i ) i ( θi )
ki
 s + θ n + α ℓ +β m  
i i i i i i

ξi

−
  s + θ n  1  s + θ n + α ℓ + β m 

i i 
i i
i i
i i 
. − 
(ti ) 
Γ


ξi
  θ i  k | ξ i | 

,


i
…(5)
Provided that
0 < Re (s + θ i n i + α i ℓ i + β i m i ) < ξ i , when ξ i > 0;
ξ i < Re (s + θ i n i + α i ℓ i + β i m i ) < 0, when ξ i < 0;
θ i ≠ 0 and n i , ( p i + γ i n i ), ( q i + δ i n i ) ∈ N 0 , where i = 1,..., n
Proof:
θ
n
Using binomial expansions for [ x i ( φ + x D x )] i , ( a i x
(ci x
βi
+ di )
qi + δi n i
αi
+ bi )
pi + γ i n i
in (2), we get
 ni
∏ h i ( x) = ∏  ∑
i =1,...,n
i =1,...,n k i =0

p i + γ i n i q i + δi n i
∑ ∑
ℓ i =0
( −1)
k i + ℓ i + mi
( − n i ) k ( −p i − γ i n i ) ℓ
mi =0
i
ki !
i
ℓi !
.
( −q i − δ i n i ) m
mi !
i
( φ)
and
n i −ki
(bi )
pi + γ i n i − ℓ i
(d i )
q i + δi n i − m i
ℓ
(a i ) i (ci )
mi
x
θi ( n i − k i )
.
The Inversion of an Integral Equation…
.
θ +1
(x i D
ki
) {x
x
αiℓ i +βi mi
e
ξ
− ti x i
43

} .

…(6)
Now applying Mellin transform of both sides of the equation (6) and using known
results [9, p.14, eq. (2.2)]; [5, p.307,eq.(7)].
We find that
 ni
∏ H i (s) = ∏  ∑
i =1,...,n
i =1,...,n k i =0

.
( −q i − δ i n i ) m
i
mi !
pi + γ i n i qi + δi n i
∑ ∑
ℓ i =0
mi =0
n i −ki
(bi )
(φ)
( −1)
k i + ℓ i + mi
pi + γ i n i − ℓ i
(d i )
( −n i ) k ( − p i − γ i n i ) ℓ
i
ki !
q i + δi n i − m i
i
ℓi !
ℓ
m
( a i ) i (c i ) i (θi )

  s + θ n 
 αiℓ i +βimi − ti x ξi

i i 

. −
, s + θ i n i 
e
 M x


θ
 


i
 k
i

ki
…(7)
Again, using [5, p.307, eq. (7)] and the known result [5, p.313, eq. (15)]
h
M {(e −ax ); s} =
1 -(s/h)  s 
a
Γ  ,
|h|
h
…(8)
We arrive at the required result (3).
3
Solution of the Integral Equation (1)
Theorem: Let the Mellin transforms Fi(s), Gi(s) and Hi(s) ≠ 0 of the functions
fi(x), gi(x) and hi(x) defined by (2) exist and be analytic in some infinite strip
ηi < Re(s) < λi of the complex s-plane. Also suppose that for a fixed σi ∈ (ηi, λi),
h *i ( x) is defined by
∏
i =1,...,n
h *i ( x) =
=
∏
i =1,...,n
i =1,...,n
H *i (s)
σ + ω∞
i
1
∏ 2πω ∫σ −ω∞ x −s H *i (s) ds,
i =1,...,n
i
Where ω = − 1 ,
∏
[M −1 {H *i (s); x}]
…(9)
44
V.B.L. Chaurasia et al.


 E
= ∏ Bi i
i =1,...,n 


 s 
Γ − 
ni
 B 

i 
∑

 k =0
s
Γ − E i −  i

Bi 

pi + γi n i q i +δi n i
∑ ∑
ℓ i =0
( −1)
k i + ℓ i + mi
mi =0
( −n i ) k ( −p i − γ i n i ) ℓ
i
ki !
ℓi !
.
.
( −q i − δ i n i ) m
mi !
i
( φ)
n i −ki
(bi )
pi + γ i n i − ℓ i
(d i )
qi + δi n i − mi
ℓi
(a i ) (ci )
mi
( θi )
ki
| ξi |
.
−1

 s + θi n i + Bi E i + C i   s + Bi E i + C i + θi n i + α i ℓ i + βi m i  
 Γ

Γ 1 +

 

θ
ξ

i


i
 ,
.

 s + B E + C + θ n + α ℓ +β m 
i i i i i i i i i

 s + θi n i + Bi E i + C i




ξi

Γ
− k i + 1 ( t i ) 



θ



i
…(10)
provided that
0 < Re (s + Bi E i + C i + θ i n i + α i ℓ i + β i m i ) < ξ i , when ξ i > 0;
ξ i < Re(s + Bi E i + C i + θ i n i + α i ℓ i + β i m i ) < 0, when ξ i < 0 ; ξ i , Bi ; θ i ≠ 0
n i , (p i + γ i n i ) , ( q i + δ i n i ) ∈ N 0 , where i = 1,..., n.
Then the integral equation (1) has its solution given by
 −BiEi −Ci ∞ −1 *  x  Bi +1

Ei
Ci


y
h
(
y
D
)
{
y
g
(
y)}
dy
∏ f i ( x) = ∏ x
,
∫0
i y
y
i
 
i =1,...,n
i =1,...,n 

…(11)
provided that the integral exists.
Proof: By convolution Theorem for Mellin transforms [5, p.308, eq.(14)],
equation (1) changes into
∏
i =1,...,n
H i (s) Fi (s) =
∏
i =1,...,n
G i (s),
…(12)
i
The Inversion of an Integral Equation…
45
Where Hi(s), Fi(s) and Gi(s) are Mellin transforms of hi(x), fi(x) and gi(x)
respectively.
Replacing s in (12) by (s+BiEi+Ci), we have
∏
i =1,...,n
Fi (s + Bi E i + C i )




 *  Ei   s + Bi E i   

= ∏ H i (s) Bi −
G i ( s + B i E i + C i ) ,


  B
i =1,...,n


i
E 




i
…(13)
Where H *i ( s) is given by (10).
Now by using known results [6, p.14, eq.(2.2)] and [5, p.307, eq. (7)], we obtain
∏
i =1,...,n
Fi (s + Bi E i + C i ) =
∏
i =1,...,n
H *i (s) [M{(y
Bi +1
D
y
)
Ei
C
y i g i ( y)}dy ; s]
…(14)
and by using the known results [5, p.307, eq. (7) and [5, p.308, eq. (14)] in (14),
we have
∏
M [x
Bi Ei +Ci
i =1,...,n
=
f i ( x); s ]
  ∞ −1 *  x  B +1

E
C
∏ M ∫0 y h i  y  ( y i D y ) i ( y i g i ( y)) dy ; s ,
 
i =1,...,n 

 
…(15)
Inverting both sides of (15) by using the Mellin inversion Theorem [5, p.307, eq.
(1)], we arrive at the required solution (11).
4
Applications
4.1. By setting i = 1 to 3 in (2), we have the following corollary
Corollary: The convolution integral equation
∏
i =1,2,3
g i ( x) =
∞
∏ ∫0
i =1,2,3
Where, the kernel
x
y −1h i   f i ( y) dy,
y
(x > 0)
…(16)
46
V.B.L. Chaurasia et al.
 ( a x αi + b ) pi ( c x βi + d ) qi
i
i
i
Kn
∏ h i ( x) = ∏  i
ξi
i
− ti x
i =1,2,3
i =1,2,3
e

ξ
p i ,q i
−ti x i 
. R n [ x, a i , b i , c i , d i ; α i , β i , γ i , δ i ; e
]
i

…(17)
ξ
 θi
ni
αi
pi + γ i n i
qi + δi n i − t i x i 
βi
= ∏ {x ( φ + x D x )} {( a i x + b i )
(ci x + d i )
e
}
i =1,2,3 

has the solution
∏
i =1,2,3
=
f i ( x)
 −BiEi −Ci ∞ −1 *  x  Bi +1

E
c
∏ x
∫0 y h i  y  ( y D y ) i {y i g i ( y)}dy ,
i =1,2,3 

provided that the integral exists and
∏
i =1,2,3
∏
i =1,2,3
…(18)
h *i ( x) is the Mellin inverse transform
H *i (s)


 E
= ∏ Bi i
i =1,2,3 


 s 
Γ − 
ni
 B 
i 

∑

 k =0
s
Γ − E i −  i

Bi 

pi + γi n i qi + δi n i
∑ ∑
ℓ i =0
( −1)
k i + ℓ i + mi
mi = 0
( −n i ) k ( −p i − γ i n i )ℓ
i
ki !
ℓi !
.
.
( −q i − δ i n i ) m
mi !
i
( φ)
n i −ki
(bi )
pi + γ i n i − ℓ i
(d i )
q i + δi n i − m i
ℓi
(a i ) (ci )
mi
( θi )
ki
| ξi |
.
−1
 s + θi n i + Bi E i + C i   s + Bi E i + C i + θi n i + α i ℓ i + βi m i  
 Γ

Γ 1 +

 

θ
ξ

i


i
 ,
.

s
+
B
E
+
C
+
θ
n
+
α
ℓ
+
β
m
 s + θi n i + Bi E i + C i

i i i i i i i i i

Γ
− k i + 1 t i
ξi



θi



…(19)
i
The Inversion of an Integral Equation…
47
provided that
0 < Re (s + Bi E i + C i + θ i n i + α i ℓ i + β i m i ) < ξ i , when ξ i > 0;
ξ i < Re(s + Bi E i + C i + θ i n i + α i ℓ i + β i m i ) < 0, when ξ i < 0 ; ξ i , Bi ; θ i ≠ 0
n i , (p i + γ i n i ) , ( q i + δ i n i ) ∈ N 0 , where i = 1,2,3.
4.2. By taking i = 1,2 in (2), the main theorem reduces to a known result recently
obtained by Chaurasia and Agnihotri in [4].
q
4.3. Since the polynomial set R p,
( x) incorporate in itself several classical as
n
well as other polynomials, solutions of a large number of convolution integral
equations for the above mentioned polynomials may be obtained by assigning
q
different values to the parameters in R p,
( x).
n
By making suitable substitution, we get the known results obtained by Srivastava
[9] and Chaurasia and Patni [3].
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]
[5]
[6]
[7]
B.D. Agrawal and J.P. Chaubey, Bilateral generating relations for a
function defined by generalized Rodrigues formula, Indian J. Pure Appl.
Math., 12(1981), 377-379.
S.K. Chatterjea, A generalization of the Bessel polynomials, Mathematics
(Cluj), 29(6) (1964a), 19-29.
V.B.L. Chaurasia and R. Patni, A class of convolution integral equations
concerning a generalized polynomial set, Soochow J. Math., 27(2001), 3743.
V.B.L. Chaurasia and M. Agnihotri, The product of generalized
polynomials sets pertaining to a class of convolution integral equlations,
Demonstratio Mathematica, XLII(3) (2009), 505-512.
A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Tables of
Integral Transforms (Vol. I), New York, McGraw-Hill, (1954).
H.M. Srivastava and R. Panda, On the unified presentation of certain
classical polynomials, Bull. Un. Mat. Ital., 12(4) (1975), 306-314.
H.M. Srivastava, K.C. Gupta and S.P. Goyal, The H-Functions of One and
Two Variables with Application, New Delhi, South Asian Publ., (1982).
48
V.B.L. Chaurasia et al.
[8]
H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions,
New York, Ellis Harwood Ltd., (1984).
R. Srivastava, The inversion of an integral equation involving a general
class of polynomials, J. Math. Anal. Appl., 186(1994), 11-20.
[9]