Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity

TL;DR

This paper introduces a new combinatorial approach to prove Touchard-Riordan-like formulas, connects q-secant numbers with Jacobi's triple product identity, and derives a finite version of this identity with applications to q-analogs of Genocchi numbers.

Contribution

It provides a novel combinatorial proof of Touchard-Riordan-like formulas and establishes a finite form of Jacobi's triple product identity linked to partition theory.

Findings

New combinatorial proof of Touchard-Riordan formulas

Finite sum version of Jacobi's triple product identity

Touchard-Riordan-like formula for q-analog of Genocchi numbers

Abstract

Touchard-Riordan-like formulas are some expressions appearing in enumeration problems and as moments of orthogonal polynomials. We begin this article with a new combinatorial approach to prove these kind of formulas, related with integer partitions. This gives a new perspective on the original result of Touchard and Riordan. But the main goal is to give a combinatorial proof of a Touchard-Riordan--like formula for q-secant numbers discovered by the first author. An interesting limit case of these objects can be directly interpreted in terms of partitions, so that we obtain a connection between the formula for q-secant numbers, and a particular case of Jacobi's triple product identity. Building on this particular case, we obtain a "finite version" of the triple product identity. It is in the form of a finite sum which is given a combinatorial meaning, so that the triple product identity can be obtained by taking the limit. Here the proof is non-combinatorial and relies on a functional equation satisfied by a T-fraction. Then from this result on the triple product identity, we derive a whole new family of Touchard-Riordan--like formulas whose combinatorics is not yet understood. Eventually, we prove a Touchard-Riordan--like formula for a q-analog of Genocchi numbers, which is related with Jacobi's identity for (q;q)^3 rather than the triple product identity.