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arXiv · 2306.10655

An Infinite Product of the Incomplete Beta Function-type Hypergeometric Function and its Probabilistic Origins

N. S. Witte
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Abstract

Recently it has been shown that the $α$-Sun density $h(x)$ [{\it J. Math. Anal. Appl.}, {\bf 527} (2023), p. 127371] which interpolates between the Fr{é}chet density and that of the positive, stable distributions whose density is given by a Fox $H$-function, has a Mellin transform involving an infinite product of ratios of Incomplete Beta functions. We develop systematic, but asymptotic, approximations for such products and consequently for the behaviour of the density as $ x\to 0+$ which complement the recent exact form for this by Simon [{\it Electron. Commun. Probab.}, {\bf 28} (2023) p. 1 - 13]. The systematic expansion is an example of a Power Product Expansion, and in our case we derive bounds and estimates which show that this expansion is not convergent and thus only yields an asymptotic expansion.

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N. S. Witte. 2023-12-02. An Infinite Product of the Incomplete Beta Function-type Hypergeometric Function and its Probabilistic Origins. https://arxiv.org/abs/2306.10655

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