Mathieu-type Series for the aleph-function Occuring in Fokker-Planck Equation

Abstract

Closed form expressions are exposed for a family of convergent Mathieu type $\mathbf a$-series and its alternating variant, whose terms contain an $\aleph$-function, which naturally occurs in certain problems associated with driftless Fokker-Planck equation with power law diffusion \cite{SBN2}. The $\aleph$-function is a generalization of the familiar $H$-function and the $I$-function. The results derived are of general character and provide an elegant generalization
for the closed form expressions the Mathieu-type series associated with the $H$--function by Pog\'any \cite{P1}, for Fox-Wright functions by Pog\'any and Srivastava \cite{PS} and for generlized hypergeometric ${}_pF_q$ and Meijer's $G$-function by Pog\'any and Tomovski \cite{PT2}, and others. For the $\overline H$-function \cite[p. 216]{MSH}
the results are obtained very recently by Pog\'any and Saxena \cite{PSax1}.