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In this article, the authors introduce the <i>q</i>-analogue of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">M</mi></semantics></math></inline-formula>-function, and establish four theorems related to the Riemann–Liouville fractional <i>q</i>-calculus operators pertaining to the newly defined <i>q</i>-analogue of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">M</mi></semantics></math></inline-formula>-functions. In addition, to establish the solution of fractional <i>q</i>-kinetic equations involving the <i>q</i>-analogue of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">M</mi></semantics></math></inline-formula>-function, we apply the technique of the <i>q</i>-Laplace transform and the <i>q</i>-Sumudu transform and its inverse to obtain the solution in closed form. Due to the general nature of the <i>q</i>-calculus operators and defined functions, a variety of results involving special functions can only be obtained by setting the parameters appropriately.