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

Ram K. Saxena, Tibor K Pogany

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}.

Keywords

$I$-function, Dirichlet series, $H$-function, Fox-Wright function, Laplace integral representation for Dirichlet series, Mathieu $\bf a$-series, Mellin-Barnes type integrals, Mittag-Leffler function

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