Filomat 27:2 (2013), 411–427
DOI 10.2298/FIL1302411A
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
On q-Sumudu Transforms of Certain q-Polynomials
Durmus¸ Albayrak
a
, Sunil Dutt Purohit
b
, Faruk Uc¸ar
a,∗
a
Department of Mathematics, Marmara University, TR-34722, Kadık¨oy, Istanbul, Turkey
b
Department of Basic Science (Mathematics), College of Technology and Engineering,
M.P. University of Agriculture and Technology Udaipur-313001 (Rajasthan), India
Abstract. Although Sumudu transform is the theoritical dual of the Laplace transform, it has many
applications in sciences and engineering for its special fundamental properties. In a previous paper [3],
we studied q-analogues of the Sumudu transform and derived some fundamental properties. This paper
follows the previous paper and aims to provide some applications of the q-Sumudu transform. The authors
give q-Sumudu transforms of some q-polynomials and q-functions. Also, we evaluated the q-Sumudu
transform of basic analogue of Fox’s H-function.
1. Introduction and Preliminaries
In the classical analysis, differential equations play a major role in mathematics, physics and engineering.
There are lots of different techniques for solving differential equations. Integral transforms were widely
used and thus a lot of work has been done on the theory and applications of integral transforms. Most
popular integral transforms are due to Laplace, Fourier, Mellin and Hankel. In 1993, the Sumudu transform
was proposed originally by Watugala [17] and he applied it to the solution of ordinary differential equations
in control engineering problems. The Sumudu transform plays a curious role in the solution of ordinary
differential equations and other branches of Mathematics and Physics. Nevertheless, this new transform
rivals the Laplace transform in problem solving. Its main advantage is the fact that it may be used to solve
problems without resorting to a new frequency domain, because it preserves scale and unit properties [4].
For further detail, one may refer to the recent papers [9]-[10] on this subject.
The theory of q-analysis, the foundation 18
th
century, in recent past have been applied in the many
areas of mathematics and physics like ordinary fractional calculus, optimal control problems, q-transform
analysis and in finding solutions of the q-difference and q-integral equations. In 1910, Jackson [7] presented
a precise definition of so-called the q-Jackson integral and developed q-calculus in a systematic way. It is
well known that, in the literature, there are two types of the q-Laplace transform and they are studied in
detail by many authors. For example [1, 6, 14].
Recently, authors [3] have introduced and study q-analogues of the Sumudu transform and derived their
fundamental properties. The aim of this paper is to give q-Sumudu transform of certain q-functions and
2010 Mathematics Subject Classification. 33D45, 33D60, 33D05, 33D15, 05A30
Keywords. q-Sumudu transforms, q-Hypergeometric functions, q-special polynomials, Fox’s H-function.
Received: 13 June 2012; Accepted: 1 November 2012
Communicated by Predrag Stanimirovic
This paper was supported by the Marmara University, Scientific Research Commission (BAPKO) under Grant 2011 FEN-A-110411-
0101.
Corresponding author
Email addresses: durmusalbayrak@marun.edu.tr (Durmus¸ Albayrak), sunil-a-purohit@yahoo.com (Sunil Dutt Purohit),
sunil_a_purohit@yahoo.com (Faruk Uc¸ar)
Durmus¸ Albayrak et al. / Filomat 27:2 (2013), 411–427 412
their special cases. For the convenience of the reader, we deem it proper to give here the basic definitions
and facts from the q-calculus.
Throughout this paper, we will assume that q satisfies the condition 0 <
q
< 1. The q-derivative D
q
f of an
arbitrary function f is given by
(D
q
f )(x) =
f (x) − f (qx)
(1 − q)x
,
where x , 0. Clearly, if f is differentiable, then
lim
q→1
−
(D
q
f )(x) =
d f (x)
dx
.
For any real number α,
[α] :=
q
α
− 1
q − 1
.
In particular, if n ∈ Z
+
, we denote
[n] =
q
n
− 1
q − 1
= q
n−1
+ ··· + q + 1.
Following usual notation are very useful in the theory of q-calculus:
a; q
n
=
n−1
k=0
1 − aq
k
,
a; q
∞
=
∞
k=0
1 − aq
k
,
(a; q)
t
=
a; q
∞
aq
t
; q
∞
( t ∈ R).
The q-analogues of the classical exponential functions are defined by
e
q
(t) =
∞
n=0
t
n
(q; q)
n
=
1
(t; q)
∞
|
t
|
< 1, (1.1)
E
q
(t) =
∞
n=0
(−1)
n
q
n(n−1)/2
t
n
(q; q)
n
= (t; q)
∞
(
t ∈ C
)
. (1.2)
By means of the (1.1) and (1.2), q-trigonometric functions defined as
sin
q
t =
e
q
(
it
)
− e
q
(
−it
)
2i
=
∞
n=0
(
−1
)
n
t
2n+1
q; q
2n+1
, (1.3)
Sin
q
t =
E
q
(
−it
)
− E
q
(
it
)
2i
=
∞
n=0
(
−1
)
n
q
n
(
2n+1
)
t
2n+1
q; q
2n+1
, (1.4)
cos
q
t =
e
q
(
it
)
+ e
q
(
−it
)
2
=
∞
n=0
(
−1
)
n
t
2n
q; q
2n
, (1.5)
Cos
q
t =
E
q
(
it
)
+ E
q
(
−it
)
2
=
∞
n=0
(
−1
)
n
q
n
(
2n−1
)
t
2n
q; q
2n
. (1.6)
Durmus¸ Albayrak et al. / Filomat 27:2 (2013), 411–427 413
Furthermore, q-hypergeometric functions are defined by
r
ϕ
s
a
1
a
2
··· a
r
b
1
b
2
··· b
s
; q, z
=
∞
n=0
(a
1
, a
2
, . . . , a
r
; q)
n
(b
1
, b
2
, . . . , b
s
; q)
n
z
n
q; q
n
,
r
ψ
s
a
1
a
2
··· a
r
b
1
b
2
··· b
s
; q, z
=
∞
n=−∞
(a
1
, a
2
, . . . , a
r
; q)
n
(b
1
, b
2
, . . . , b
s
; q)
n
(
−1
)
n
q
n
2
s−r
z
n
,
and
m−k
Φ
m−1
a
1
a
2
··· a
m−k
b
1
b
2
··· b
m−1
; q, z
=
∞
n=0
(a
1
, a
2
, . . . , a
m−k
; q)
n
(b
1
, b
2
, . . . , b
m−1
; q)
n
(
−1
)
n
q
n
2
k
z
n
q; q
n
.
where
(a
1
, a
2
, . . . , a
r
; q)
n
=
r
i=1
a
i
; q
n
.
For further detail and properties about q-hypergeometric functions see [5, 12, 13] and many others.
If a function f (x) has a series expansion as follow
f
(
x
)
=
∞
n=−∞
a
n
x
n
,
then the following function is well defined
f
x − y
=
∞
n=−∞
a
n
x − y
n
q
. (1.7)
The improper integral is defined by [7, 11]
x
0
f (t)d
q
t = x(1 − q)
∞
k=0
q
k
f (xq
k
), (1.8)
∞/A
0
f (x)d
q
x = (1 − q)
k∈Z
q
k
A
f (
q
k
A
). (1.9)
q-analogues of gamma and beta functions defined as follow [11]
Γ
q
(
α
)
=
1/
(
1−q
)
0
x
α−1
E
q
q
1 − q
x
d
q
x
(
α > 0
)
, (1.10)
Γ
q
(
α
)
= K
(
A; α
)
∞/A
(
1−q
)
0
x
α−1
e
q
−
1 − q
x
d
q
x
(
α > 0
)
, (1.11)
where
K
(
A; t
)
= A
t−1
−q/A; q
∞
−q
t
/A; q
∞
−A; q
∞
−Aq
1−t
; q
∞
(
t ∈ R
)
. (1.12)
The function K
(
A; t
)
provides the following equation for the variable t (see [11]):
K(x; t + 1) = q
t
K(x; t) (1.13)
Durmus¸ Albayrak et al. / Filomat 27:2 (2013), 411–427 414
q-gamma function has the following property
Γ
q
(
x
)
=
q; q
∞
q
x
; q
∞
1 − q
1−x
=
q; q
x−1
1 − q
x−1
, (1.14)
and we have
lim
q→1
−
Γ
q
(
α
)
= Γ
(
α
)
.
q-Bessel functions were introduced by Jackson [8] in 1905 and are therefore referred to as Jackson’s q-Bessel
functions. Some q-analogues of the Bessel functions are given by
J
(
1
)
ν
z; q
=
q
ν+1
; q
∞
q; q
∞
z
2
ν
2
Φ
1
0 0
q
ν+1
; q, −
z
2
4
,
|
z
|
< 2
=
z
2
ν
∞
n=0
−z
2
/4
n
q; q
ν+n
q; q
n
(1.15)
and
J
(
2
)
ν
z; q
=
q
ν+1
; q
∞
q; q
∞
z
2
ν
0
Φ
1
−
q
ν+1
; q, −
q
ν+1
z
2
4
=
z
2
ν
∞
n=0
q
n
(
n+ν
)
−z
2
/4
n
q; q
ν+n
q; q
n
. (1.16)
q
-Bessel functions are related by the following equality
J
(
2
)
ν
z; q
=
−
z
2
4
; q
∞
J
(
1
)
ν
z; q
,
|
z
|
< 2.
q-Bessel functions are q-extensions of the Bessel function of the first kind since
lim
q→1
−
J
(
k
)
ν
1 − q
z; q
= J
ν
(
z
)
, k = 1, 2.
The third kind q-analogue of the Bessel function is given by following formula
J
(
3
)
ν
z; q
=
q
ν+1
; q
∞
q; q
∞
z
ν
1
Φ
1
0
q
ν+1
; q, qz
2
= z
ν
∞
n=0
(
−1
)
n
q
n
(
n−1
)
/2
qz
2
n
q; q
ν+n
q; q
n
. (1.17)
This third kind q-Bessel function is also known as the Hahn-Exton q-Bessel function. This is also q-extension
of the Bessel function of the first kind since
lim
q→1
−
J
(
3
)
ν
1 − q
z; q
= J
ν
(
2z
)
.
Albayrak, Purohit and Uc¸ar [3] defined the q-analogues of the Sumudu transform by means of the following
q-integrals
S
q
{f (t); s} =
1
1 − q
s
s
0
E
q
q
s
t
f
(
t
)
d
q
t, s ∈
(
−τ
1
, τ
2
)
, (1.18)
Durmus¸ Albayrak et al. / Filomat 27:2 (2013), 411–427 415
over the set of functions
A =
f (t)|∃M, τ
1
, τ
2
> 0, |f (t)| < ME
q
|
t
|
/τ
j
, t ∈ (−1)
j
× [0, ∞)
and
S
q
{f (t); s} =
1
1 − q
s
∞
0
e
q
−
1
s
t
f
(
t
)
d
q
t, s ∈
(
−τ
1
, τ
2
)
, (1.19)
over the set of functions
B =
f (t)|∃M, τ
1
, τ
2
> 0, |f (t)| < Me
q
|
t
|
/τ
j
, t ∈ (−1)
j
× [0, ∞)
.
By virtue of (1.8) and (1.9), q-Sumudu transforms can be expressed as
S
q
f
(
t
)
; s
= (q; q)
∞
∞
k=0
q
k
f
sq
k
(q; q)
k
, (1.20)
and
S
q
f
(
t
)
; s
=
s
−1
−
1
s
; q
∞
k∈Z
q
k
f (q
k
)
−
1
s
; q
k
. (1.21)
2. Main Theorems
In this section, we shall investigate certain theorems, which gives a number of image formulas involving
q-hypergeometric functions, under the q-Sumudu transforms.
Theorem 2.1. In correspondence to the bounded sequence A
n
, let f (x) is given by
f
(
x
)
=
∞
n=0
A
n
x
n
, (2.1)
then for α > 0 the following results hold:
S
q
x
α−1
f
(
x
)
; s
= s
α−1
1 − q
α−1
∞
n=0
A
n
Γ
q
(
α + n
)
1 − q
s
n
, (2.2)
S
q
x
α−1
f
(
x
)
; s
= s
α−1
1 − q
α−1
∞
n=0
A
n
Γ
q
(
α + n
)
K
(
s; α + n
)
1 − q
s
n
. (2.3)
Proof. In view of (1.20) and (2 .1), we have
S
q
x
α−1
f
(
x
)
; s
= (q; q)
∞
∞
k=0
q
k
sq
k
α−1
(q; q)
k
f
sq
k
= (q; q)
∞
∞
k=0
q
k
sq
k
α−1
(q; q)
k
∞
n=0
A
n
sq
k
n
= s
α−1
(q; q)
∞
∞
n=0
A
n
s
n
∞
k=0
q
α+n
k
(q; q)
k
. (2.4)