Contiguous Relations, Laplace's Methods and Continued Fractions for 3F2(1)

TL;DR

This paper develops new continued fraction expansions for ratios of hypergeometric series 3F2(1) using contiguous relations and Laplace's methods, providing precise error estimates and demonstrating rapid convergence.

Contribution

It introduces a novel approach combining contiguous relations with discrete Laplace's method to generate and analyze continued fractions for hypergeometric series ratios.

Findings

Constructed infinite continued fraction expansions for 3F2(1) ratios

Established exact error bounds for approximants

Proved rapid convergence of the continued fractions

Abstract

Using contiguous relations we construct an infinite number of continued fraction expansions for ratios of generalized hypergeometric series 3F2(1). We establish exact error term estimates for their approximants and prove their rapid convergences. To do so we develop a discrete version of Laplace's method for hypergeometric series in addition to the use of ordinary (continuous) Laplace's method for Euler's hypergeometric integrals.