On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs

Abstract

We give a direct and simple proof of Touchard's continued fraction, provide an extension of it, and transform it into similar expansions related to Motzkin and Schroeder numbers. Another proof is then given that uses only induction. We use this machinery on two examples that appear in recent papers of Josuat-Verges; with an additional parameter, these two can be treated simultaneously.