Continued fraction formulae involving ratios of three gamma functions

TL;DR

This paper derives seven new continued fraction formulas involving ratios of three gamma functions, extending Ramanujan's work, and rigorously proves five of them using the Bauer-Muir transformation twice.

Contribution

It introduces seven novel continued fraction formulas with gamma functions and provides rigorous proofs for five using a novel application of the Bauer-Muir transformation.

Findings

Seven new continued fraction formulas involving gamma functions.

Five formulas rigorously proved using Bauer-Muir transformation.

Extension of Ramanujan's Entry 34 in Chapter 12.

Abstract

Via the MC-algorithm, in this paper we produce seven continued fraction formulae involving products and quotients of three gamma functions with three parameters, and another is an extension of Entry 34 in Chapter 12 of Ramanujan's second notebook. Five of them will be proved rigorously by the Bauer-Muir transformation. A crucial ingredient in the proofs of our five theorems is to employ the Bauer-Muir transformation twice with different nonlinear modifying factors.