R E S E A R C H
Open Access
Some results for Laplace-type integral
operator in quantum calculus
Shrideh K.Q. Al-Omari
1, Dumitru Baleanu
2,3and Sunil D. Purohit
4**Correspondence: [email protected] 4Department of HEAS
(Mathematics), Rajasthan Technical University, Kota, India
Full list of author information is available at the end of the article
Abstract
In the present article, we wish to discussq-analogues of Laplace-type integrals on diverse types ofq-special functions involving Fox’sHq-functions. Some of the discussed functions are theq-Bessel functions of the first kind, theq-Bessel functions of the second kind, theq-Bessel functions of the third kind, and theq-Struve functions as well. Also, we obtain some associated results related toq-analogues of the
Laplace-type integral on hyperbolic sine (cosine) functions and some others of exponential order type as an application to the given theory.
Keywords: Jv(x;q) function;Yv(x;q) function;Kv(x;q) function;Hv(x;q) function; Laplace-type integral
1 Introduction and preliminaries
Quantum calculus is a version of calculus where derivatives are differences and antideriva-tives are sums, and no further limits are required. The quantum calculus orq-calculus, compared to the differential and integral calculus, has been very recently named. Hence some rules and definitions need to be recalled. For 0 <q< 1, theq-calculus starts with the definition of theq-analogue of the differential and theq-analogue of derivatives as well. Theq-analogue of the integern, the factorial ofn, and the binomial coefficient are respectively given as
[n]q=
1 –qn
1 –q,
[n]q
! =
n
1[k]q, n∈N
1, n= 0
,
n k
q
=
n
1
1 –qn–k+1
1 –qk . (1)
Theq-analogue of (x+a)n(n∈N) and itsq-derivative are respectively given as
(x+a)nq=
n–1
j=0
x+qja, Dq(x+a)nq= [n]q(x+a)nq–1, (x+a)0q= 1. (2)
Theq-Jackson integrals from 0 toaand fromatobare given as follows (see [1], see also [2]):
a
0
f(x)dqx= (1 –q)a
∞
0
faqkqk (3)
and
a b
f(x)dqx=
b
0
f(x)dqx–
a
0
f(x)dqx. (4)
The improperq-Jackson integral is given as follows (see [1]):
∞
A
0
f(x)dqx= (1 –q)
n∈Z qk
Af
qk
A
, A∈C.
Theq-analogues of the gamma function are defined by
q(α) =
1
1–q
0
xα–1Eq
q(1 –q)xdqx
and
q(α) =K(A;α)
∞
A(1–q)
0
xα–1eq
–(1 –q)xdqx,
whereα> 0 and, for everyt∈R,
K(A;t) =At–1(–q/A;q)∞
(–qt/A;q)∞
(–A;q)∞
(–Aq1–t;q)∞.
Here
(a;q)n= n–1
0
1 –aqk, (a;q)∞= lim
n→ ∞(a;q)n.
The very useful identities used in this article are (cf. [2])
q(x) =
(q;q)∞
(qx;q)∞(1 –q)
1–x and (a;q) t=
(a;q)∞
(aqt;q)∞, t∈R.
Theq-hypergeometric functions are represented by
rφs
a1,a2, . . . ,ar
α1,α2, . . . ,αs
q,z
=
∞
0
(a1,a2, . . . ,ar;q)n
(α1,α2, . . . ,αs;q)n
zn
(q;q)n
and
m–km–1
a1,a2, . . . ,am–k
α1,α2, . . . ,αm–1
q,z
=
∞
0
(a1, . . . ,am–k;q)n
(α1, . . . ,αm–1;q)n
(–1)nq(n2)k
× zn (q;q)n
,
where (a1,a2, . . . ,ap;q)n=
p
2 H-Function and related functions
TheH-function, which is an extension of the hypergeometric functionspFq, introduced
by Fox [3] (see also [4, 5]), has found various applications in a huge range of problems asso-ciated with reaction, reaction diffusion, communication, engineering, fractional differen-tial equations, integral equations, theoretical physics, and statistical distribution theory as well. TheH-functions have also been recognized to play a fundamental role in frac-tional calculus with its applications. Fox’sH-function, admitting to a standard notation, is presented as
Hpm,q,n(η) = 1 2πi
P
jpm,q,n(w)ηwdw, (5)
wherePis a suitable complex path,ηw=exp{w(log|η|+iargη)},jpm,q,n(w) = AC((ws))DB((ww)), and
A(w) =
m
1
(bj–βjw), B(w) = n
1
(1 –aj+αjw),
C(w) =
q m+1
(1 –bj–βjw), D(w) = p n+1
(aj+αjw),
0≤n≤p, 1≤m≤q, {aj,bj} ∈C, {αj,βj} ∈R+. Letαjandβjbe positive integers and
0≤m≤N; 0≤n≤M. Then theq-analogue of Fox’sH-function is given as (see [6])
HMm,,nN
x;q
(a1,α1), (a2,α2), . . . , (aμ,αM)
(b1,β1), (b2,β2), . . . , (bN,βN)
= 1
2πi
C
m j=1G(qbj–
βjs)n
j=1G(q1–aj+ αjs)πxs
N
j=m+1G(q1–bj+ βjs)M
j=n+1G(qaj–
αjs)G(q1–s)sinπsdqs,
whereGis defined in terms of the product
Gqα=
∞
k=0
1 –qα–k–1= 1 (qα;q)
∞. (6)
The contourCis parallel toRe(ws) = 0, such that all poles ofG(qbj–βjs), 1≤j≤m, are its
right and those ofG(q1–aj+αjs), 1≤j≤n, are the left ofC. The above integral converges if
Re(slogx–log sinπs) < 0, for huge values of|s|onC. Hence,
arg(x) –w2w–11 log|x|<π, |q|< 1, logq= –w= –w1–iw2,
wherew1andw2are real numbers.
Indeed, forαi=βj= 1, for alli,j, we write theq-analogue of Meijer’sG-function as
GmM,,nN
x;q
a1,a1, . . . ,aM
b1,b2, . . . ,bN
= 1
2πi
C
m
j=1G(qbj–s)
n
j=1G(q1–aj+s)πx2
N
j=m+1G(q1–bj+s)
M
j=n+1G(qaj–s)G(q1–s)sinπs
dqs, (7)
Additionally, theq-analogues of the Bessel functionJv(x) of the first kind, the Bessel
function ofYv(x), the Bessel function of the third kindKv(x), and Struve’s functionHv(x)
are, respectively, defined in terms of Fox’sHq-function by [7] as follows:
Jv(x;q) =
In [8] (see also [9]), someq-analogues of the natural exponential functions, sine func-tions, cosine funcfunc-tions, hyperbolic sine funcfunc-tions, and hyperbolic cosine functions are, respectively, given in terms of FoxsH-function as follows:
eq(–x) =G(q)H0,21,0
On the other hand, some impressive integral transforms also have the correspondingq -analogues in the concept ofq-calculus; they include theq-Laplace transforms [10], theq -Sumudu transforms [9, 11–13], theq-Wavelet transform [14], theq-Mellin transform [15],
so on. Recently, a number of authors have studied various image formulas for theseq -integral transforms, associated with a variety of special functions. In this sequel, we aim to investigate theq-analogues of Laplace-type integrals on diverse types ofq-special func-tions involving Fox’sHq-function.
3 q-Laplace-type transforms forHq-function
A Laplace-type integral was introduced in [20, 21]. Theq-analogues of the Laplace-type integral of the first kind were defined later by [22] as follows:
qL2
f(ξ);y= 1 1 –q2
y–1
0
ξEq2
q2y2ξ2f(ξ)dξ
=(q
2;q2)
∞
[2]qy2
∞
i=0 q2i
(q2;q2)
i
fqiy–1, (17)
whereas theq-analogues of the Laplace-type integral of the second kind were defined by
q2
f(ξ);y= 1 1 –q2
∞
0
ξeq2
y2ξ2dqξ
= 1
[2]q(–y2;q2)∞
i∈Z
q2ifqi–y2;q2i. (18)
For the sake of convenience, we establish some formulas for theqL2operator. A similar
argument can give certain corresponding results for the operatorq2.
Theorem 1 Letβbe a positive real number.Then
qL2
ξ2β–2(y) = (q
2;q2)
∞
[2]qy2(qβ;q2)∞
.
Proof By using (17), we have
qL2
ξ2β–2;y=(q
2;q2)
∞
[2]qy2β
∞
i=0 q2i
(q2;q2)
i
qiy–12β–2
=(q
2;q2)
∞
[2]qy2β
∞
i=0
q2βiy2β–2
(q2;q2)
i
.
That is,
qL2
ξ2β–2;y=(q
2;q2)
∞
[2]qy2
∞
i=0 q2βi
(q2;q2)
i
. (19)
By the fact that
eq(z) =
∞
i=0 zi
(q;q)i
we have
This completes the establishment of the belief.
Theorem 2 Letλbe a complex number.Then
qL2
Proof Letλbe a complex number. Then by (17) we obtain
qL2
By invoking (21) in (20), we get
qL2
By inserting the identity
in (22) yields
Now, on account of the definition ofHq-function, we may establish that
qL2
providedk< 0.
The proof is completed.
4 Applications to trigonometric and hyperbolic functions
In this part, we shall give certain natural relevance to the leading results.
Theorem 3 Let eqbe defined in terms of(12).Then
The demonstration of this theorem is finished.
Theorem 4 Letsinqbe defined in terms of(13).Then we have
qL2
Proof The proof of this theorem indeed follows from substituting the valuesλ= 0,k= 1, andγ=(1–4q2)2 and from multiplying by√π(1 –q2)–12 {G(q2)}2.
Theorem 5 Letcosqbe defined in terms of(14).Then
The proof is completed.
Theorem 6 Letsinhqbe defined in terms of(15).Then
qL2
The proof is completed.
Theorem 7 Letcoshqbe defined in terms of(16).Then
qL2
Proof The validation of this theorem is identical to that of the previous theorem.
Theorem 8 Let the Bessel function be defined in terms of(8).Then
qL2
Theorem 9 Let the q-Bessel function of the second kind be defined in terms of(9)–(11).
qL2
Kv
x;q2(y) =(1 –q
2)
[2]qy4
q2;q2∞
×H1,32,1
(1 –q2)2
4y2 ;q 2
(0, 1)
(2v, 1), (–2v, 1), (1, 1)
,
qL2
Hv
x;q2(y) = (1 –q
2)1–α
21–α[2]
qy4
q2;q2∞
×H2,43,2
(1 –q2)2 4y2 ;q
2
(0, 1), (1–2α, 1) (v2, 1), (–2v, 1), (1+α
2 , 1)(1, 1)
.
Proof Proof of this theorem follows from (9)–(11) and the technique quite similar to that
of Theorems 3–8. We omit the details.
Acknowledgements
The authors are thankful to the referee for his/her valuable remarks and comments for the improvement of the paper.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Author details
1Department of Basic Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman, Jordan. 2Department of Mathematics, Cankaya University, Ankara, Turkey.3Institute of Space Sciences, Magurele-Bucharest, Romania. 4Department of HEAS (Mathematics), Rajasthan Technical University, Kota, India.
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Received: 13 February 2018 Accepted: 16 March 2018
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