# Continued fraction expansions of the generating functions of Bernoulli and related numbers

## TL;DR

This paper derives continued fraction expansions for generating functions of Bernoulli, Cauchy, Euler, and harmonic numbers, providing explicit convergents and exploring transformations, thus advancing understanding of their mathematical structure.

## Contribution

It introduces new continued fraction expansions for various special number sequences and discusses their transformations, enriching the analytical tools for these functions.

## Key findings

- Explicit continued fraction forms for Bernoulli, Cauchy, Euler, and harmonic numbers.
- Derivation of convergents for these continued fractions.
- Analysis of linear fractional transformations of the continued fractions.

## Abstract

We give continued fraction expansions of the generating functions of Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, and their generalized or related numbers. In particular, we focus on explicit forms of the convergents of these continued fraction expansions. Linear fractional transformations of such continued fractions are also discussed. We show more continued fraction expansion for different numbers and types, in particular, on Cauchy numbers.