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arXiv · 2403.05708

Extending the Meijer $G$-function

Dmitrii Karp·Alexey Kuznetsov
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Abstract

By replacing the Euler gamma function by the Barnes double gamma function in the definition of the Meijer $G$-function, we introduce a new family of special functions, which we call $K$-functions. This is a very general class of functions, which includes as special cases Meijer $G$-functions (thus also all hypergeometric functions ${}_p F_q$) as well as several new functions that appeared recently in the literature. Our goal is to define the $K$-function, study its analytic and transformation properties and relate it to several functions that appeared recently in the study of random processes and the fractional Laplacian. We further introduce a generalization of the Kilbas-Saigo function and show that it is a special case of $K$-function.

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Dmitrii Karp, Alexey Kuznetsov. 2024-03-08. Extending the Meijer $G$-function. https://arxiv.org/abs/2403.05708

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