Addition Theorems Via Continued Fractions

TL;DR

This paper explores addition theorems related to special functions like Bessel and hypergeometric functions, using continued fractions and connections to classical theorems, with applications to Hankel determinants.

Contribution

It introduces new addition formulas for special functions via continued fractions and links them to classical theorems, expanding the theoretical framework.

Findings

Derived new addition theorems for Bessel functions

Established addition formulas for hypergeometric functions

Applied results to evaluate Hankel determinants

Abstract

We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for Bessel functions and confluent hypergeometric functions. We also derive several additions theorems for basic hypergeometric functions. Applications to the evaluation of Hankel determinants are also given .