On the q-Laplace Transform and Related Special Functions

Abstract

Motivated by statistical mechanics contexts, we study the properties of the q-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships between q-exponential and other well-known functional forms, such as Mittag-Leffler functions, hypergeometric and H-function, by means of the kernel function of the integral. Traditionally, we have been applying the Laplace transform method to solve differential equations and boundary value problems. Here, we propose an alternative, the q-Laplace transform method, to solve differential equations, such as as the fractional space-time diffusion equation, the generalized kinetic equation and the time fractional heat equation.

A β -Convolution Theorem Associated with the General Quantum Difference Operator

“…See [1][2][3][4] for some applications of the convolution in electrical engineering, physics, and theory of distribution. The classical convolution theorem says that the Laplace transform of the convolution u * v of the two functions u and v is equal to the Laplace transform of u multiplied by the Laplace transform of v. Recently, there are many versions of the convolution theorem such as h-convolution theorem, q-convolution theorem, convolution theorem on time scale, and ðq, hÞ-convolution theorem on discrete time scale, see, e.g., [5][6][7][8]. Bohner and Guseinov [9] studied the convolution theorem on a time scale T , where the convolution of two functions u, v : T ⟶ ℝ is defined by…”

“…where Δ is the delta differentiation and σ is the forward jump operator in T . In [7], the convolution theorem of two positive real functions u 1 ðtÞ and u 2 ðtÞ is defined by…”

“…It should be noted that, here in [7], the Laplace transform denoted by L q and the related convolution are defined by the usual integral; however, the q-exponential e −st q is used in the form ½1 + ðq − 1Þst −1/ðq−1Þ , t ≥ 0, q > 1. In [10], the q -convolution is defined by…”

On a system of q‐modified Laplace transform and its applications

“…Among them the study has done on q‐Laplace transform 28–31 . We start from the definition of the Laplace transform (Deakin 32 ) of the function f ( α ) the inverse of the Laplace transform is as follows In Purohit and Kalla, 29 Naik and Haubold, 33 and Chung et al, 34 using q‐Jackson integral (Jackson 22 ) the q‐analogue of the Laplace transform given by …”

“…e px {𝑓 (𝛽)}d𝛽, (c ≥ 0). In Purohit and Kalla, 29 Naik and Haubold, 33 and Chung et al, 34 using q-Jackson integral (Jackson 22 ) the q-analogue of the Laplace transform given by q L 𝛾 {𝑓 (𝜗)} =…”