Combinatorics of Continuants of Continued Fractions with 3 Limits

TL;DR

This paper explores the combinatorial structure of continuants in continued fractions with three diverging limits, deriving polynomial identities and applications to counting sequences.

Contribution

It provides new combinatorial descriptions for continuants with three limits and links them to classical identities, expanding understanding of continued fractions.

Findings

Derived polynomial identities from combinatorial descriptions

Connected continuants to classical combinatorial identities

Applied results to counting sequences

Abstract

We give combinatorial descriptions of the terms occurring in continuants of general continued fractions that diverge to three limits. Equating these with the usual combinatorial descriptions due to Euler, Sylvester, and Minding induces nontrivial polynomial identities. Special cases and applications to counting sequences are given.