Random continued fractions with beta hypergeometric distribution

TL;DR

This paper generalizes the class of random continued fractions with beta-hypergeometric distributions using Thomae's formula, involving graph-based constructions and a detailed parameter analysis rooted in classical results.

Contribution

It introduces a novel framework for constructing random continued fractions with explicit beta-hypergeometric densities via graph-based methods and classical mathematical results.

Findings

Extended the class of random continued fractions with explicit beta-hypergeometric densities

Developed seven graph structures leading to new random continued fractions

Analyzed parameter conditions for beta-hypergeometric distributions using classical results

Abstract

In a recent paper (Asci \textit{et al.}, 2008) it has been shown that certain random continued fractions have a density which is a product of a beta density and a hypergeometric function $_{2}F_{1}$. In the present paper we fully exploit a formula due to Thomae (1879) in order to generalize substantially the class of random continuous fractions with a density of the above form. This involves the design of seven particular graphs. Infinite paths on them lead to random continued fractions with an explicit distribution. A careful study about the set of five real parameters leading to a beta-hypergeometric distribution is required, relying on almost forgotten results mainly due to Felix Klein.