q-Analogues and properties of the Laplace-type integral operator in the quantum calculus theory

In this paper, we discuss the q-Laplace-type integral operator on certain class of special functions. We propose q-analogues and obtain results involving polynomials of even orders and functions of q-trigonometric types. Moreover, we establish results related to q-hyperbolic functions and certain q-differential operators. Relying on the given q-differentiation formulas, we finally derive the nth derivative of the q-Laplace-type integral and attain formulas including q-convolution products.


Introduction
The subject of quantum calculus is known as the calculus without limits; it has earned an eminent reputation and publicity due to its vast applications in mathematics, statistics and physics. Much of the theory of the quantum calculus relies on the Jackson qderivatives and q-integrals, which replace the classical derivative by a difference operator, which allows one to deal with sets of non-differentiable functions. Recently, this area has been stimulated to grow rapidly by many researchers and a variety of new results can be found in Refs. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references cited therein. By fixing a real number q such that 0 < q < 1, the q-derivative of a differentiable function ϑ is defined by [1] The q-integrals from 0 to y and from 0 to ∞ are, respectively, defined by [1] y 0 ϑ(x) d q x = (1q)y ∞ n=0 ϑ yq n q n (1) and provided their respective series converge absolutely. The q-integration by parts is given by [16] b a g( In the literature, there have been found two types of q-analogues of the exponential function introduced as [16][17][18] and where [n] q = q n-1 + · · · + q + 1 and [n] q ! = [n] q · · · [n] q are the q-analogues of the integer and its factorial, respectively. The q-analogues of the gamma function are also given by [5] Γ q (t) = The useful properties of the q-gamma functions Γ q andΓ q are obtained in the literature as follows.
Theorem 1 Let n ∈ N and t ∈ R. Then the following identities hold: The q-analogues of the trigonometric functions sin x and cos x are given by (see, e.g., [4]) This paper is organized as follows. In Sect. 1, we present some preliminaries and notations that are very useful in the sequel. In Sect. 2, we apply the q-analogues of the Laplacetype integral operator to certain polynomials and functions of special-types. In Sect. 3, we apply the q-analogue F 2,q to a certain class of differential operators of arbitrary order. In Sect. 4, we discuss the convolution product and establish a convolution theorem of the F 2,q integral operator.
2 F 2,q and S 2,q analogues of some q-special functions The Laplace integral operator, among various integral operators, is the most popular and widely used in several branches of engineering sciences and applied mathematics. The Laplace-type integral has been firstly defined by Yürekli and Sadek [19] and extended to a space of generalized functions by Al-Omari [20]. In [4], Ucar et al. have given the qanalogues of the Laplace-type integral operator of some elementary functions by making a free use of the identities of the q-Laplace integral operator. One of the goals of this paper is to provide the q-analogues of the Laplace-type integral operator, with a different approach, and derive results involving elementary functions and some other difference operators. The Laplace-type integral operator [19, (1.4)] has a close relation with the familiar Laplace integral operator given as The q-analogues of the Laplace-type integral operator were introduced by [4] q L 2 ϑ(x); y = 1 1q 2 and q l 2 ϑ(x); y = provided Re(y) > 0. Here, we introduce two q-analogues of the Laplace-type integral operator in the standard way as and provided Re(y) > 0. In what follows, we make a use of Eq. (9) and Eq. (10) and provide a summary of some results related to the F 2,q and S 2,q analogues in the course of the following theorems.
Theorem 2 Let δ be a real number. Then the following hold: Proof By aid of Eq. (9), we proceed by Therefore, by the change of variables x 2 y 2 = z we obtain Hence, the proof of the first part (i) of the theorem follows from Eq. (6). The proof of the second part (ii) follows by similar techniques. The proof of the theorem is completed.
Consequently, we state without proof the following straightforward corollary.

Corollary 3
The following identities hold: Theorem 4 Let δ be a positive real number. Then the following identities hold: Proof Let δ be a positive real number. Then, by employing Eq. (9) and Eq. (5) we write By using the change of variables z = x 2 y 2 , the above equation yields Hence, by invoking the definition of the gamma function given by Eq. (6), we obtain Therefore, by applying Theorem 1, Eq. (11) leads to the geometric series This proves the first part of the theorem. By following a like approach, we get This, indeed, proves the second part of the theorem. The proof of the third part (iii) follows from a similar technique. To establish the fourth part (iv) of the theorem, we make use of the definition of the q-analogue E q to write By applying the change of variables z = x 2 y 2 , our previous equation becomes Hence, by using Part (iv) of Theorem 1, the equation can be put into the form Therefore, simple computations yield Hence, the proof of this theorem is therefore completed.
Proof Let δ be a positive real number. Then, by the definition of F 2,q and the definition of Cos q , we write By making the change of variables z = x 2 y 2 we derive Consequently, by employing Theorem 1, Eq. (12) turns out to be in the form of the geometric series Therefore, the convergence condition of the geometric series shows that F 2,q Cos q δx 2 ; y = y 2 [2] q (y 2 + δ 2 ) δ 2 < y 4 .
The proof of Part (i) is therefore finished. The proof of the Part (ii) follows from Theorem 1, Part (v) and a similar argument to that employed in the first part. Once again, an argument similar to the argument we have employed for Part (i) establishes Part (iii). Finally, Part (iv) is a straightforward result of Theorem 2. This finishes the proof of the theorem.
A similar statement to the statement of Theorem 5 can be read as follows.

Theorem 6
Let δ be a positive real number. Then the following identities hold true: The proof of this theorem follows from definitions and a similar argument to that we already checked for Theorem 5. Details are therefore omitted. Now, by following the usual notations of [16] and the facts that cosh q x = e q (x) + e q (-x) 2 and sinh q x = e q (x)e q (-x) 2 (13) we state the following straightforward corollary.

Corollary 7
Let δ be positive real number. Then we have .
The proof of this corollary directly follows from Eq. (13), Theorems 5 and 6. Details are omitted.

F 2,q of q-differential operators
We devote this section to computations related to the F 2,q integral and some differential operators. First of all, we derive the following theorem.
But simple computations then yield Hence, shifting the lower bound of the above summation implies The proof of the theorem is therefore finished.
Proof By the q-integration by parts given by Eq. (3), we write Hence, by applying Theorem 8, Eq. (15) can be expressed as Therefore, by changing the variables as qx = z (d q x = 1 q d q z), we transfer Eq. (16) into the form Thus, by multiplying the previous equation by q -n q n , we obtain [n] q ! q -3n y 2n q n z 2n d q z [n] q ! q n+1 q -3n y 2n q n z 2n d q z.
Thus, this equation can be nicely expressed as Finally, the preceding equation can be written as This finishes the proof of the theorem.

Theorem 10 Let y be a positive real number. Then we have
Proof Let y > 0. Then, by using Theorem 9, we obtain Hence, the proof of the theorem is finished.

Theorem 11 Let y be a positive real number. Then we have
Proof By taking into account Theorem 9 and using simple computations, we have By utilizing Theorem 10, Eq. (18) together with computations reveals This finishes the proof of the theorem. Proof Let y > 0 be given arbitrary. Then, by employing Theorem 10, we derive that Therefore, further simplifications on the above equation yield Hence, the proof of the theorem is finished.
By following techniques similar to the techniques already used for Theorems 10-12, we reach the following result. Let us now check the following differentiation formula.

Theorem 14 Let y be a positive real number. Then we have
Proof Let y be a positive real number. Then, from the definitions, we have But, as earlier, we know that Hence, invoking these results in the previous summation gives This finishes the proof of the theorem.

Theorem 15
Let q (x) = 1 x D x,q be given. Then we have Proof By making use of Eq. (9) and employing Eq. (3), we establish that Therefore, substituting the integral bounds in the first part suggests we have Hence, by virtue of Theorem 10, Eq. (19) reveals The change of variables qx = z yields d q x = 1 q d q z. Therefore, the above equation together with certain technical computations implies Hence, the proof of this theorem is completed.

F 2,q of q-convolution products
In this section we focus on giving a convolution theorem for the F 2,q integral operator. For, let us assume ϑ(t) = t 2δ and θ (t) = t 2γ -1 , δ, γ > 0. Then the q-convolution product is defined for ϑ and θ as (see [15,Eq. (44) provided the integral exists. The following is a very useful property.
This finishes the proof of the above result. Now, by the aid of Theorem 16, we establish the following convolution theorem.

Conclusion
The q-Laplace-type integral operator was applied to certain polynomials and various functions of q-trigonometric types. Such results were extended to q-hyperbolic functions and some other q-differential operators. By using q-differentiation formulas, several derivatives of the q-Laplace-type integral operator were obtained. On top of that, some related formulas as well as a convolution theorem were also discussed in detail.