Ratios of the Gauss hypergeometric functions with parameters shifted by integers: part I

TL;DR

This paper derives explicit formulas and integral representations for ratios of Gauss hypergeometric functions with shifted parameters, analyzing their properties and connections to Nevanlinna classes and continued fractions.

Contribution

It provides new formulas for hypergeometric function ratios, their jump behavior, and their classification within generalized Nevanlinna classes, expanding understanding of their analytic structure.

Findings

Explicit formulas for the jump of the ratio over the branch cut.

Integral representations for ratios with mild asymptotic behavior.

Connections between Nevanlinna classes, continued fractions, and hypergeometric functions.

Abstract

We consider the ratio of two Gauss hypergeometric functions with real parameters shifted by arbitrary integers. We find a formula for the jump of this ratio over the branch cut in terms of a real hypergeometric polynomial, the beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral representations for such ratio when the asymptotic behaviour at unity is mild and the denominator does not vanish. Multiplying the ratio by the aforementioned polynomial, we obtain a function that belongs to a generalized Nevanlinna class for arbitrary values of real parameters. We give an in-depth analysis of a particular case known as the Gauss ratio. Furthermore, we establish a few general facts relating generalized Nevanlinna classes to Jacobi and Stieltjes continued fractions, as well as to factorization formulae for these classes. The results are illustrated with a large number of examples.