Abstract
In the present article, we wish to discuss q-analogues of Laplace-type integrals on diverse types of q-special functions involving Fox’s \(H_{q}\)-functions. Some of the discussed functions are the q-Bessel functions of the first kind, the q-Bessel functions of the second kind, the q-Bessel functions of the third kind, and the q-Struve functions as well. Also, we obtain some associated results related to q-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.
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1 Introduction and preliminaries
Quantum calculus is a version of calculus where derivatives are differences and antiderivatives are sums, and no further limits are required. The quantum calculus or q-calculus, compared to the differential and integral calculus, has been very recently named. Hence some rules and definitions need to be recalled. For \(0< q<1\), the q-calculus starts with the definition of the q-analogue of the differential and the q-analogue of derivatives as well. The q-analogue of the integer n, the factorial of n, and the binomial coefficient are respectively given as
The q-analogue of \(( x+a ) ^{n}\) (\(n\in \mathbb{N} \)) and its q-derivative are respectively given as
The q-Jackson integrals from 0 to a and from a to b are given as follows (see [1], see also [2]):
and
The improper q-Jackson integral is given as follows (see [1]):
The q-analogues of the gamma function are defined by
and
where \(\alpha >0\) and, for every \(t\in \mathbb{R}\),
Here
The very useful identities used in this article are (cf. [2])
The q-hypergeometric functions are represented by
and
where \(( a_{1},a_{2},\ldots,a_{p};q ) _{n}=\prod_{k=0} ^{p} ( a_{k};q ) _{n}\).
2 H-Function and related functions
The H-function, which is an extension of the hypergeometric functions \(_{p}F_{q}\), introduced by Fox [3] (see also [4, 5]), has found various applications in a huge range of problems associated with reaction, reaction diffusion, communication, engineering, fractional differential equations, integral equations, theoretical physics, and statistical distribution theory as well. The H-functions have also been recognized to play a fundamental role in fractional calculus with its applications. Fox’s H-function, admitting to a standard notation, is presented as
where P is a suitable complex path, \(\eta^{w}=\exp \{ w ( \log \vert \eta \vert +i\arg \eta ) \} \), \(\jmath_{p,q}^{m,n} ( w ) =\frac{A ( w ) B ( w ) }{C ( s ) D ( w ) }\), and
\(0\leq n\leq p\), \(1\leq m\leq q\), \(\{ a_{j},b_{j} \} \in \mathbb{C} \), \(\{ \alpha_{j},\beta_{j} \} \in \mathbb{R} ^{+}\). Let \(\alpha_{j}\) and \(\beta_{j}\) be positive integers and \(0\leq m\leq N\); \(0\leq n\leq M\). Then the q-analogue of Fox’s H-function is given as (see [6])
where G is defined in terms of the product
The contour C is parallel to \(\operatorname{Re} ( ws ) =0\), such that all poles of \(G ( q^{b_{j}-\beta_{j}s} ) \), \(1\leq j\leq m\), are its right and those of \(G ( q^{1-a_{j}+\alpha_{j}s} ) \), \(1\leq j\leq n\), are the left of C. The above integral converges if \(\operatorname{Re} ( s\log x-\log \sin \pi s ) <0\), for huge values of \(\vert s \vert \) on C. Hence,
where \(w_{1}\) and \(w_{2}\) are real numbers.
Indeed, for \(\alpha_{i}=\beta_{j}=1\), for all i, j, we write the q-analogue of Meijer’s G-function as
where \(0\leq m\leq N\); \(0\leq n\leq M\) and \(\operatorname{Re} ( s\log x-\log \sin \pi s ) <0\).
Additionally, the q-analogues of the Bessel function \(J_{v} ( x ) \) of the first kind, the Bessel function of \(Y_{v} ( x ) \), the Bessel function of the third kind \(K_{v} ( x ) \), and Struve’s function \(H_{v} ( x ) \) are, respectively, defined in terms of Fox’s \(H_{q}\)-function by [7] as follows:
In [8] (see also [9]), some q-analogues of the natural exponential functions, sine functions, cosine functions, hyperbolic sine functions, and hyperbolic cosine functions are, respectively, given in terms of Fox′s H-function as follows:
On the other hand, some impressive integral transforms also have the corresponding q-analogues in the concept of q-calculus; they include the q-Laplace transforms [10], the q-Sumudu transforms [9, 11–13], the q-Wavelet transform [14], the q-Mellin transform [15], q-\(E_{2,1}\)-transform [16], q-Mangontarum transforms [17, 18], q-natural transforms [19], and so on. Recently, a number of authors have studied various image formulas for these q-integral transforms, associated with a variety of special functions. In this sequel, we aim to investigate the q-analogues of Laplace-type integrals on diverse types of q-special functions involving Fox’s \(H_{q}\)-function.
3 q-Laplace-type transforms for \(H_{q}\)-function
A Laplace-type integral was introduced in [20, 21]. The q-analogues of the Laplace-type integral of the first kind were defined later by [22] as follows:
whereas the q-analogues of the Laplace-type integral of the second kind were defined by
For the sake of convenience, we establish some formulas for the \(_{q}L_{2}\) operator. A similar argument can give certain corresponding results for the operator \(_{q}\ell_{2}\).
Theorem 1
Let β be a positive real number. Then
Proof
By using (17), we have
That is,
By the fact that
we have
This completes the establishment of the belief. □
Theorem 2
Let λ be a complex number. Then
where \(0\leq n\leq m\) and \(0\leq m\leq N\) and λ is an arbitrary complex number.
Proof
Let λ be a complex number. Then by (17) we obtain
Let \(\beta =\lambda +k z+1\), then by Theorem 1 we have
By invoking (21) in (20), we get
By inserting the identity
in (22) yields
Now, on account of the definition of \(H_{q}\)-function, we may establish that
provided \(k<0\).
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 \(e_{q}\) be defined in terms of (12). Then
Proof
By setting \(\lambda =0\), \(\gamma =1-q^{2}\), and \(k=1\), Theorem 3 immediately follows from Theorem 2. □
The demonstration of this theorem is finished.
Theorem 4
Let \(\sin_{q}\) be defined in terms of (13). Then we have
Proof
The proof of this theorem indeed follows from substituting the values \(\lambda =0\), \(k=1\), and \(\gamma = \frac{ ( 1-q^{2} ) ^{2}}{4} \) and from multiplying by \(\sqrt{\pi } ( 1-q^{2} ) ^{\frac{-1}{2}} \{ G ( q ^{2} ) \} ^{2}\).
Hence, the proof is completed. □
Theorem 5
Let \(\cos_{q}\) be defined in terms of (14). Then
Proof
Proof follows from Theorem 2 for \(\lambda =0\), \(k=1\), \(\gamma =\frac{ ( 1-q^{2} ) ^{2}}{4}\).
The proof is completed. □
Theorem 6
Let \(\sinh_{q}\) be defined in terms of (15). Then
Proof
By using the special case, \(\lambda =0\), \(k=1\), \(\gamma =\frac{ ( 1-q ^{2} ) ^{2}}{4}\).
The proof is completed. □
Theorem 7
Let \(\cosh_{q}\) be defined in terms of (16). Then
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
Proof
By setting \(\lambda =0\), \(k=1\), \(\gamma =\frac{1-q^{2}}{4}\) and multiplying by \(\{ G ( q^{2} ) \} ^{2}\), the result follows. □
Theorem 9
Let the q-Bessel function of the second kind be defined in terms of (9)–(11). Then
Proof
Proof of this theorem follows from (9)–(11) and the technique quite similar to that of Theorems 3–8. We omit the details. □
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Al-Omari, S.K.Q., Baleanu, D. & Purohit, S.D. Some results for Laplace-type integral operator in quantum calculus. Adv Differ Equ 2018, 124 (2018). https://doi.org/10.1186/s13662-018-1567-1
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DOI: https://doi.org/10.1186/s13662-018-1567-1