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Recently it has been shown that the $\alpha$-Sun density $h(x)$ [{\it J.Math. Anal. Appl.}, {\bf 527} (2023), p. 127371] which interpolates between theFr{\'e}chet density and that of the positive, stable distributions whosedensity is given by a Fox $H$-function, has a Mellin transform involving aninfinite product of ratios of Incomplete Beta functions. We develop systematic,but asymptotic, approximations for such products and consequently for thebehaviour of the density as $ x\to 0+$ which complement the recent exact formfor 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 ourcase we derive bounds and estimates which show that this expansion is notconvergent and thus only yields an asymptotic expansion.

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