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

Abstract

Touchard–Riordan-like formulas are certain expressions appearing in enumeration problems and as moments of orthogonal polynomials. We begin this article with a new combinatorial approach to prove such formulas, related to 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.