Fractional <i>q</i>-Calculus Operators Pertaining to the <i>q</i>-Analogue of <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">M</mi></mrow></semantics></math></inline-formula>-Function and Its Application to Fractional <i>q</i>-Kinetic Equations

Abstract

In this article, the authors introduce the q-analogue of the M-function, and establish four theorems related to the Riemann–Liouville fractional q-calculus operators pertaining to the newly defined q-analogue of M-functions. In addition, to establish the solution of fractional q-kinetic equations involving the q-analogue of the M-function, we apply the technique of the q-Laplace transform and the q-Sumudu transform and its inverse to obtain the solution in closed form. Due to the general nature of the q-calculus operators and defined functions, a variety of results involving special functions can only be obtained by setting the parameters appropriately.

Keywords

• fractional <i>q</i>-calculus operators
• fractional <i>q</i>-kinetic equations
• <i>q</i>-Mittag–Leffler function
• <i>q</i>-Laplace transform
• <i>q</i>-Sumudu transform
• <i>q</i>-gamma function