Evaluating complex inverse formulas for q-Sumudu transforms
DURMUŞ ALBAYRAK1FARUK UÇAR1SUNIL DUTT PUROHIT2,3
1Department of Mathematics, Marmara University, 34722, Kadıköy, Istanbul, T85.(<
2Department of HEAS (Mathematics), Rajasthan Technical University, 324010, Kota, INDIA
3Dept. of Computer Science and Mathematics, Lebanese American University, 13-5053, Beirut, LEBANON
Abstract: In this paper, q-analogues of the Sumudu transform, along with an inversion formula and some explicit
computations, are presented. This work essentially focuses on q-analogues of the inverse Sumudu transform and
the construction method of the inversion formula via a path integral along a Bromwich contour. It is also shown
how the complex inversion formulas considered in this paper admit q-expansions that yield various inverse q-
Sumudu transforms for q-series.
Key-Words: Integral transforms, Sumudu transform, q-Sumudu transforms, Complex inversion formula,
Received: May 12, 2022. Revised: August 13, 2023. Accepted: September 15, 2023. Available online: November 16, 2023.
1 Introduction
Integral transforms have gained importance and are
ubiquitous, mainly because of their tremendous abil-
ity to be used in various fields of applied sciences and
engineering. The best known and mostly used inte-
gral transforms are Laplace, Fourier, Mellin and Han-
kel. In 1993, Watugala, [1], added another dimension
to this research by proposing a new integral transform,
which is called the Sumudu transform, and he used
it in control engineering problems to obtain the solu-
tions of certain ordinary differential equations. In this
way, Weerakoon, [2], provide the Sumudu transform
of partial derivatives as well as the complex inversion
formula for this transform, and put it to use in the solu-
tion of partial differential equations. Despite the fact
that the Sumudu transform is the hypothetical dual of
the Laplace transform, it has a wide range of applica-
tions in science and engineering due to its special core
properties. Its main advantage is that it can be used to
solve problems without having to use a new frequency
domain, since it has scale and unit conservation prop-
erties [3]. Readers are recommended to refer to [4],
[5], [6] for more information on this matter.
The origin of q-calculus dates back to the late 18th
century. In some works, q-calculus is also referred to
as limitless calculus. The letter qcomes from quan-
tum. In recent years, q-calculus has found its applica-
tions in many fields, especially quantum mechanics.
In q-calculus the concepts of q-series, q-derivatives
and q-integrals have as much importance as series,
derivatives and integrals in classical calculus [7]. The
q-series has been applied in numerous areas of mathe-
matics and physics, such as optimal control problems,
[8], arbitrary order (fractional) computation, [9], q-
transform analysis, [10], geometric function theory,
[11], and the discovery of solutions to the q-difference
equations [12], [13]. q-differential (or q-difference)
equations arise as a result of mathematical modeling
in the solution of many problems in mathematics and
physics, just like the classical ones. One of the most
comprehensive techniques for solving q-differential
(or q-difference) equations is the q-integral transform
method. In this method, the most commonly used
q-integral transforms to solved the q-differential (or
q-difference) equations are q-Laplace, q-Fourier, q-
Mellin transforms. (see, [14], [15], [16], [17], [18],
[19], [20], [21], [22], [23], [24])
Through this method, a q-differential (or q-
difference) equation is reduced to an algebraic prob-
lem that is easier to solve with the help of the trans-
formed function, rather than the q-derivative. The in-
verse q-integral transform is then used to obtain the
solution of the original problem. Motivated by the
applications of classical Sumudu transform, Albayrak
et al. considered and studied basic (or q-) analogues
of Sumudu transform. They also considered basic
properties and gave q-Sumudu transforms of some q-
functions and their special cases (see [25, 26]). Cer-
tain inversion and representation theorems and their
applications to q-Sumudu transforms are also dis-
cussed in [27]. More recently, Purohit and Uçar, [28],
using the q-Sumudu transforms gave an alternative
solution to the q-kinetic equation with fractional q-
integral operators of the Riemann-Liouville category.
Though it is acceptable to rely on lists and tables
of function transforms while looking for solutions to
differential equations, having access to inverse trans-
form formulas is always a significantly better tool for
engineers and mathematicians. Using the Cauchy hy-
pothesis together with the residue theorem, Belgacem
and Karaballi [4] proved a theorem (using Bromwich
contour integral) with respect to the complex inverse
Sumudu transform. In this article, we aim to present
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certain complex inversion formulas for the q-Sumudu
transforms. Our main result is a q-extension of the
consequence of Belgacem and Karaballi [4] and can
be used to derive certain intriguing consequences and
exceptional cases.
2 Preliminaries
To deal with this work effectively, we use some ba-
sics of quantum theory. For more subtleties of the q-
calculus, one can refer to [29]. Throughout this paper,
we will consider qas a particular quantity satisfying
the agreement 0<|q|<1.
The q-derivative of a given function φis defined
by
(Dqφ)(x) = φ(x)−φ(qx)
(1 −q)x.(1)
This definition holds when xis not equal to zero.
If the function φis differentiable, it is clear that as
the parameter qapproaches 1 from the left, the q-
derivative converges to the classical derivative.
The q-shifted factorial is defined by
(a;q)n=
1, n = 0
n
Q
k=0 1−aqk, n = 1,2, ... (2)
and has properties as follows:
(a;q)∞=
∞
Y
n=0
(1 −aqn),(3)
(a;q)α=(a;q)∞
(aqα;q)∞
,(4)
1
q;1
qn
= (−1)nq−(n+1
2)(q;q)n.(5)
where a= 0,|q|<1, α ∈R. We recall that the
basic analogues of the classical exponential function
are defined by
eq(t) =
∞
X
n=0
tn
(q;q)n
=1
(t;q)∞
,(6)
where |t|<1and
Eq(t) =
∞
X
n=0
qn(n−1)/2(−t)n
(q;q)n
= (t;q)∞,(7)
where t∈Cand the relationship between eq(t)and
Eq(t)is as follows:
Eq(qt) = e1/q(t).(8)
By virtue of the (6) and (7),q-trigonometric functions
can be defined as follows [29]:
sinqt=eq(it)−eq(−it)
2i
=
∞
X
n=0
(−1)nt2n+1
(q;q)2n+1
,(9)
Sin qt=Eq(−it)−Eq(it)
2i
=
∞
X
n=0
(−1)nqn(2n+1) t2n+1
(q;q)2n+1
,(10)
cosqt=eq(it) + eq(−it)
2
=
∞
X
n=0
(−1)nt2n
(q;q)2n
,(11)
Cosqt=Eq(it) + Eq(−it)
2
=
∞
X
n=0
(−1)nqn(2n−1) t2n
(q;q)2n
.(12)
Note that, classic exponential function etand trigono-
metric functions sin tand cos tcan be obtained as the
limit case (q→1−) of eq((1 −q)t), Eq((q−1) t),
sinq((1 −q)t),Sin q((q−1) t),cosq((1 −q)t),
Cosq((q−1) t).
A basic hypergeometric series is defined by the
following summation [29, p.4]:
rϕs(a1, a2, . . . , ar;b1, b2, . . . , bs;q, z)
≡rϕsa1, a2, . . . , ar
b1, b2, . . . , bs;q, z
=
∞
X
n=0
(a1;q)n(a2;q)n. . . (ar;q)n
(b1;q)n(b1;q)n. . . (bs;q)n(−1)nq(n
2)s−r+1 zn
(q;q)n
where q= 0, r > s+1. Jackson, [30], introduced the
following q-analogues of Bessel function and there-
fore they are referred as Jackson’s q-Bessel functions
(or basic Bessel function) in some papers:
J(1)
ν(z;q) = qν+1;q∞
(q;q)∞z
2ν
2Φ10 0
qν+1 ;q, −z2
4,|z|<2
=z
2ν∞
X
n=0
(−z/2)2n
(q;q)ν+n(q;q)n
,(13)
and
J(2)
ν(z;q) = qν+1;q∞
(q;q)∞z
2ν
0Φ1−
qν+1 ;q, −qν+1z2
4
=z
2ν∞
P
n=0
qn(n+ν)(−z/2)2n
(q;q)ν+n(q;q)n
(14)
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where the q-shifted factorials are defined as in (2).
The simple association among q-Bessel functions is
as
J(2)
ν(z;q) = −z2
4;q∞
J(1)
ν(z;q),|z|<2.
The q-Bessel functions mentioned above can be con-
sidered as q-extensions of the first kind Bessel func-
tion. The third kind q-analogue of the Bessel function,
some authors refer to this function as Hahn-Exton q-
Bessel function, is given by the subsequent formula
J(3)
ν(z;q) = qν+1;q∞
(q;q)∞
zν1Φ10
qν+1 ;q, qz2
=zν∞
P
n=0
(−1)nqn(n−1)/2 qz2n
(q;q)ν+n(q;q)n
.(15)
Here we note that, in the limiting cases for q→1−
we obtain
lim
q→1−
J(k)
ν((1 −q)z;q) = Jν(z) (k= 1,2)
and
lim
q→1−
J(3)
ν((1 −q)z;q) = Jν(2z).
Jackson, [31], introduced q-definite integrals, namely
the Jackson integral is defined by
Zx
0
f(t)dqt=x(1 −q)
∞
X
k=0
qkf(xqk),(16)
Z∞/A
0
f(x)dqx= (1 −q)X
k∈Z
qk
Afqk
A.(17)
Hahn, [18], defined the q-analogues of the well-
known classical Laplace transform by means of the
following q-integrals
F1(s) = Lq{f(t); s}
=1
1−qZs−1
0
Eq(qst)f(t)dqt, (18)
and
F2(s) = Lq{f(t); s}
=1
1−qZ∞
0
eq(−st)f(t)dqt, (19)
where Re (s)>0and q-analogues of the classical
exponential functions are defined by (6) and (8).Al-
bayrak et al. [25] characterized q-analogues of the
Sumudu transform via the succeeding q-integrals
G1(s) = Sq{f(t); s}
=1
(1 −q)sZs
0
Eqq
stf(t)dqt, (20)
where s∈(−τ1, τ2)for the set of functions
A={f(t)|∃M, τ1, τ2>0,|f(t)|< MEq(|t|/τj),
t∈(−1)j×[0,∞),
and
G2(s) = Sq{f(t); s}
=1
(1 −q)sZ∞
0
eq−1
stf(t)dqt, (21)
where s∈(−τ1, τ2)considering the set of functions
B={f(t)|∃M, τ1, τ2>0,|f(t)|< Meq(|t|/τj),
t∈(−1)j×[0,∞).
As a result of (16) and (17),q-Laplace and q-Sumudu
transforms can be asserted as
Lq{f(t); s}=(q;q)∞
s
∞
X
k=0
qkfs−1qk
(q;q)k
,(22)
Sq{f(t) ; s}= (q;q)∞
∞
X
k=0
qkfsqk
(q;q)k
,(23)
and
Lq{f(t); s}
=1
(−s;q)∞
∞
X
k=0
qkfqk(−s;q)k,(24)
Sq{f(t) ; s}
=s−1
−1
s;q∞X
k∈Z
qkf(qk)−1
s;qk
.(25)
In [18], [25] the following limit cases were proved:
lim
q→1−
Lq{f(t) ; (1 −q)s}=lim
q→1−
Lq{f(t) ; (q−1) s}
=L{f(t) ; s}
and
lim
q→1−
Sq{f(t) ; (1 −q)s}=lim
q→1−
Sq{f(t) ; (q−1) s}
=S{f(t) ; s}
where
F0(s) = L{f(t); s}=Z∞
0
e−stf(t)dt, Re s > 0,
and
G0(s) = S{f(t); s}=1
sZ∞
0
e−t/sf(t)dt, s ∈(−τ1, τ2),
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over the set of functions
A=nf(t)∃M, τ1, τ2>0,|f(t)|< M e|t|/τj,
t∈(−1)j×[0,∞).
The relationship between classical Laplace and
Sumudu transform and the relationship between their
q-versions for i= 0,1,2can be written as follows:
Gi(s) = 1
sFi1
sor Fi(s) = 1
sGi1
s.
In [4] the authors proved a theorem for defining
the inverse Sumudu transform using the Cauchy the-
orem, the residue theorem, and the Bromwich con-
tour as shown in Figure 1. However, they used the
relationship between classical Laplace and Sumudu
transforms, as well as the definition of the classi-
cal inverse Laplace transform, to establish inverse
Sumudu transform. In our work, we did not need to q-
versions of this relationship or the definition of the q-
inverse Laplace transform to find q-variants of inverse
Sumudu transform. By leveraging the following the-
orems, the Bromwich contour illustrated in Figure 1,
and the definitions of the q-Sumudu transforms, we
have successfully defined the q-inverse transforms.
3 Complex Inversion Formula
Though it is acceptable to rely on lists and tables of
function transforms while looking for solutions to dif-
ferential equations, having access to inverse trans-
form formulas is always a significantly better tool for
engineers and mathematicians. In this section, we will
give complex inversion formulas for q-Sumudu trans-
forms, starting with q-version of the results given in
[4]. The following theorem gives us the complex in-
verse q-Sumudu transforms, derived from the Cauchy
and residue theorems, using a Bromwich contour.
Theorem 1. Let Sq{f(t) ; s}=G1(s)and
lim
s→∞ G1(s) = 0.Assume that G1(s) /sis a meromor-
phic function with singularities in the region where
Re (s)< c. If a circular domain γexists with radius
Rand positive constants, Mand k, with
G1(s)
s
< MR−k,
then f(t)is given by
S−1
q{G1(s) ; t}=1
2πi Zc+i∞
c−i∞
eqt
sG1(s)
sds
=1
(q;q)∞
∞
X
n=0
(−1)nq(n+1
2)
(q, q)n
G1(tqn).(26)
Figure 1: Bromwich contour
Proof of Theorem 1.Let s∈Cand integration takes
place in the complex plane along the line s=c.
To show this and from the conditions of the theo-
rem, c, which is the abscissa of point C, is chosen
to replace the real part of all singularities of G1(1
s)
s,
including branch points, principal singularities and
poles. For G1(1
s)
s, one can compute the integral over
the Bromwich contour as shown in Figure 1 to accom-
modate all cases, including infinitely many singular-
ities. Therefore, without loss of generality, assuming
that the singularities G1(1
s)
sare poles, that they all lie
to the left of the real line s=c.
Let us start by evaluating the following integral
along a path described in Figure 1:
S−1
q{G1(s) ; t}=1
2πi ZAB+γ
eqt
sG1(s)
sds
=1
2πi ZAB
+ZBDE
+ZEF
+ZF JK
+ZKL
+ZLNAeqt
sG1(s)
sds.
Along AB: T=√R2−c2and G1(s) /sis ana-
lytic on AB and for sufficently large values of R, the
following is imply
lim
R→∞
1
2πi ZAB
eqt
sG1(s)
sds
=lim
R→∞
1
2πi Zc+iT
c−iT
eqt
sG1(s)
sds
=1
2πi Zc+i∞
c−i∞
eqt
sG1(s)
sds
=XRes eqt
sG1(s)
s.
From the definition of (6), we have
XRes eqt
sG1(s)
s=XRes "1
t
s;q∞
G1(s)
s#.
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Since 1
(t
s;q)∞
G1(s)
shas poles at sn=tqn
(n= 0,1,2, ...), we get
XRes "1
t
s;q∞
G1(s)
s#
=
∞
X
n=0
lim
s→sn (s−tqn)
∞
Y
k=0
1
1−t
sqk
G1(s)
s!
=
∞
X
n=0
lim
s→sn
∞
Y
k=0
k=n
1
1−t
sqkG1(s)
=
∞
X
n=0
∞
Y
k=0
k=n1
(1 −qk−n)G1(tqn)
=1
(q;q)∞
∞
X
n=0
(−1)nq(n+1
2)
(q, q)n
G1(tqn).
Along BDE and LNA: We can find some parame-
ters M > 0, k > 0such that on BDE and LNA
G1(s)
s
< MR−k,
then the integral along BDE and LNA of
eqt
sG1(s) /sapproaches zero as R→ ∞
lim
R→∞
1
2πi ZBDE
eqt
sG1(s)
sds = 0,
lim
R→∞
1
2πi ZLNA
eqt
sG1(s)
sds = 0.
Along EF: s=xeπi, ds =eπidx and in the pro-
cess of smoves from −Rto −r,xgoes from Rto
r:
lim
R→∞
r→0
1
2πi ZEF
eqt
sG1(s)
sds
=lim
R→∞
r→0
1
2πi Zr
R
eq−t
xG1(−x)
xdx.
Along FJK: s=reiθ, ds =ireiθdθ, and since
lim
s→∞ G1(s) = 0,we have for sufficently small r
lim
r→0
1
2πi ZF JK
eqt
sG1(s)
sds
=lim
r→0
1
2πi Z−π
π
eqt
reiθ G1reiθdθ = 0.
Along KL: s=xe−πi, ds =e−πidx and as sgoes
from −rto −R,xgoes from rto R:
lim
R→∞
r→0
1
2πi ZKL
eqt
sG1(s)
sds
=lim
R→∞
r→0
1
2πi ZR
r
eq−t
xG1(−x)
xdx.
Thus we obtain
S−1
q{G1(s) ; t}=1
2πi Zc+i∞
c−i∞
eqt
sG1(s)
sds
=1
(q;q)∞
∞
X
n=0
(−1)nq(n+1
2)
(q, q)n
G1(tqn).
Theorem 2. Let Sq{f(t) ; s}=G2(s)and its value
zero at ∞.Assume that G2(s) /sis a meromor-
phic function with singularities in the region where
Re (s)< c. If a circular domain γexists with radius
Rand positive constants, Mand k, with
G2(s)
s
< MR−k,
then f(t)is given by
S−1
q{G2(s) ; t}=1
2πi Zc+i∞
c−i∞
Eq−qt
sG2(s)
sds
=1
1
q;1
q∞
∞
X
n=0
1
(q, q)n
G2−t
qn.(27)
Proof of Theorem 2.The theorem is proved in the
same way and under the same assumptions as in the
preceding one. We evaluate integrals along the same
path described in previous theorem:
S−1
q{G2(s) ; t}=1
2πi ZAB+γ
Eq−qt
sG2(s)
sds
=1
2πi ZAB
+ZBDE
+ZEF
+ZF JK
+ZKL
+ZLNAEq−qt
sG2(s)
sds.
Along AB: T=√R2−c2and G2(s) /sis ana-
lytic on AB and for sufficently large values of R, we
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have
lim
R→∞
1
2πi ZAB
Eq−qt
sG2(s)
sds
=lim
R→∞
1
2πi Zc+iT
c−iT
Eq−qt
sG2(s)
sds
=1
2πi Zc+i∞
c−i∞
Eq−qt
sG2(s)
sds
=XRes Eq−qt
sG2(s)
s.
Making use of (8) and (6) we have
XRes Eq−qt
sG2(s)
s
=XRes e1/q−t
sG2(s)
s
=XRes
1
−t
s;1
q∞
G2(s)
s
Since 1
(−t
s;1
q)∞
G2(s)
shas poles at sn=−t
qn
(n= 0,1,2, ...), we get
XRes
1
−t
s;1
q∞
G2(s)
s
=
∞
X
n=0
lim
s→sn
s+t
qn∞
Y
k=0
1
1 + t
sqk
G2(s)
s
=
∞
X
n=0
lim
s→sn
∞
Y
k=0
k=n
1
1 + t
sqkG2(s)
=
∞
X
n=0
∞
Y
k=0
k=n1
(1 −qn−k)G2−t
qn
=1
1
q;1
q∞
∞
X
n=0
1
(q, q)n
G2−t
qn.
Along BDE and LNA: We can find constans M >
0, k > 0such that on BDE and LNA
G2(s)
s
< MR−k,
then the integral along BDE and LNA of
Eq−qt
sG2(s)
sapproaches zero as R→ ∞
lim
R→∞
1
2πi ZBDE
Eq−qt
sG2(s)
sds = 0,
lim
R→∞
1
2πi ZLNA
Eq−qt
sG2(s)
sds = 0.
Along EF: s=xeπi, ds =eπidx and as sgoes
from −Rto −r,xgoes from Rto r:
lim
R→∞
r→0
1
2πi ZEF
Eq−qt
sG2(s)
sds
=lim
R→∞
r→0
1
2πi Zr
R
Eqqt
xG2(−x)
xdx.
Along FJK: s=reiθ, ds =rieiθ,and since
lim
s→∞ G2(s) = 0,for sufficently small r, we have
lim
r→0
1
2πi ZF JK
Eq−qt
sG2(s)
sds
=lim
r→0
1
2πi Z−π
π
Eq−qt
reiθ G2reiθdθ = 0.
Along KL: s=xe−πi, ds =e−πidx and as sgoes
from −rto −R,xgoes from rto R:
lim
R→∞
r→0
1
2πi ZKL
Eq−qt
sG2(s)
sds
=lim
R→∞
r→0
1
2πi ZR
r
Eqqt
xG2(−x)
xdx.
Thus, we obtain
S−1
q{G2(s) ; t}=1
2πi Zc+i∞
c−i∞
Eq−qt
sG2(s)
sds
=1
1
q;1
q∞
∞
X
n=0
1
(q;q)n
G2−t
qn.
4 Applications
In the present section, we will give some examples
and consider inverse q-Sumudu transforms of the q-
series by the help of main theorems. Throughout this
section, we will assume that G1(s) = Sq{f(t) ; s}
and G2(s) = Sq{f(t) ; s}.
Example 1. Let
Gi(s) = sj(∀j∈Z, i = 1,2) .
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Let us consider S−1
q{G1(s) ; t}first. Using (26) we
obtain
S−1
q{G1(s) ; t}=1
(q;q)∞
∞
X
n=0
(−1)nq(n+1
2)
(q, q)n
G1(tqn)
=tj
(q;q)∞
∞
X
n=0
(−1)nq(n
2)
(q, q)nqj+1n.
Hence by (7)and (4), we have
S−1
q{G1(s) ; t}=tjqj+1;q∞
(q;q)∞
=tj
(q;q)j
.
Now, we consider S−1
q{G2(s) ; t}. Similarly, as in
the previous one, using (27) we obtain
S−1
q{G2(s) ; t}=1
1
q;1
q∞
∞
X
n=0
1
(q, q)n
G2−t
qn
=(−1)jtj
1
q;1
q∞
∞
X
n=0
(−1)nq−(n+1
2)
1
q;1
qnq−jn.
Making use of (7),(4) and (5), we have
S−1
q{G2(s) ; t}=
(−1)jtj1
qj+1 ;1
q∞
1
q;1
q∞
=(−1)jtj
1
q;1
qj
=q(j+1
2)tj
(q;q)j
.
Consequently, we obtain
S−1
qsj;t=1
(q;q)j
tj,(28)
S−1
qsj;t=q(j+1
2)
(q;q)j
tj.(29)
Now, we consider an interesting application of
the main results, which provides the inverse q-
Sumudu transforms of q-series functions. Further-
more, we also provide a comprehensive list of inverse
q-Sumudu transforms for certain functions available
in the literature.
Example 2. If Gi(s) (i= 1,2) has in the form
Gi(s) =
∞
X
j=0
ajsaj+b,i= 1,2,
aj+1
aj
<1, a, b ∈Z
then
S−1
q{G1(s) ; t}=
∞
X
j=0
aj
taj+b
(q;q)aj+b
,(30)
S−1
q{G2(s) ; t}=
∞
X
j=0
aj
q(aj+b+1
2)taj+b
(q;q)aj+b
.(31)
To show (30), we consider S−1
q{G1(s) ; t}. From
(26), we have
S−1
q{G1(s) ; t}
=1
(q;q)∞
∞
X
n=0
(−1)nq(n+1
2)
(q, q)n
G1(tqn)
=1
(q;q)∞
∞
X
n=0
(−1)nq(n+1
2)
(q, q)n
∞
X
j=0
aj(tqn)aj+b
=
∞
X
j=0
aj
taj+b
(q;q)∞
∞
X
n=0
(−1)nq(n
2)
(q, q)nqaj+b+1n.
Using (7) and (4), we get
S−1
q{G1(s) ; t}=
∞
X
j=0
aj
taj+bqaj+b+1;q∞
(q;q)∞
=
∞
X
j=0
aj
taj+b
(q;q)aj+b
.
Now, we consider S−1
q{G2(s) ; t}. Similarly, by
(27), we have
S−1
q{G2(s) ; t}
=1
1
q;1
q∞
∞
X
n=0
1
(q, q)n
G2−t
qn
=1
1
q;1
q∞
∞
X
n=0
1
(q, q)n
∞
X
j=0
aj−t
qnaj+b
=
∞
X
j=0
aj
(−1)aj+btaj+b
1
q;1
q∞
∞
X
n=0
(−1)nq−(n
2)
1
q;1
qnq−aj−b−1n.
Making use of (7),(4) and (5), we obtain
S−1
q{G2(s) ; t}
=
∞
X
j=0
aj
(−1)aj+btaj+b1
qaj+b+1 ;1
q∞
1
q;1
q∞
=
∞
X
j=0
aj
(−1)aj+btaj+b
1
q;1
qaj+b
=
∞
X
j=0
aj
q(aj+b+1
2)taj+b
(q;q)aj+b
.
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Example 3. If we set aj=(−1)jaj
(q;q)j
,a= 1, b = 0 in
the previous example, we have
Gi(s) =
∞
X
j=0
ajsaj+b=
∞
X
j=0
(−1)jaj
(q;q)j
sj=eq(−as)
for i= 1,2. Thus, using the results (30) and (31), we
get
S−1
q{eq(−as); t}=
∞
X
j=0
(−1)jaj
(q;q)j
tj
(q;q)j
=J(1)
02√at;q,
S−1
q{eq(−as); t}=
∞
X
j=0
(−1)jaj
(q;q)j
q(j+1
2)tj
(q;q)j
=J(3)
02√at;q.
We have summarized the results obtained with
some specific choices of ajin Examples 2 and 3 in
Table 1 (Appendix) and Table 2 (Appendix).
5 Concluding Remark
In this paper we focused on evaluating complex inver-
sion formulas for the q-Sumudu transforms. The spe-
cial cases of (30) and (31) can be found in Table 1 (Ap-
pendix) and Table 2 (Appendix). As a final remark
it can be said that, the complex inversion formulas,
deduced in the previous section for q-Sumudu trans-
forms, are significant and can yield numerous inverse
q-Sumudu transforms for variety of q-functions. The
results obtained in this paper provide q-extensions of
the outcome given in, [4], as mentioned earlier. q-
difference equations are q-functional equations which
relate q-functions with their q-derivatives. We think
that as an application of the our results obtained here,
finding the solutions of q-difference equations such as
non-homogenous q-difference equations with linear
constant coefficients, non-homogenous q-difference
equations with linear variable coefficients will be the
main subject of our future studies.
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Appendix
Table 1: Special cases of (30)
aja b G1(s)f(t) = S−1
q{G1(s) ; t}
aj1 0 1
1−as eq(at)
(−1)ja2j+1 2 1 as
1 + a2s2sinq(at)
(−1)ja2j2 0 1
1 + a2s2cosq(at)
(−1)jajq(j
2)1 0 1Φ1(q; 0; q;as)Eq(at)
(−1)jqj(2j+1)a2j+1 2 1 as1Φ1q4; 0; q4;a2s2q3Sin q(at)
(−1)jqj(2j−1)a2j2 0 1Φ1q4; 0; q4;a2s2qCosq(at)
(−1)jaj
(q;q)j
1 0 eq(−as)J(1)
02√at;q
Table 2: Special cases of (31)
aja b G2(s)f(t) = S−1
q{G2(s) ; t}
−a
qj
1 0 q
q+as Eq(at)
(−1)ja2j+1
q2j+1 2 1 qas
q2+a2s2Sin q(at)
(−1)ja2j
q2j2 0 q2
q2+a2s2Cosq(at)
(−1)jaj
(q;q)j
1 0 eq(−as)J(3)
0√at;q
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