# Touchard-Riordan formulas, T-fractions, and Jacobis triple product

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1 Fakultät für Mathematik 2 UMN-MATH - School of Mathematics

Abstract : Touchard-Riordan formulas are some expressions appearing in enumeration prob- lems and as moments of orthogonal polynomials. We begin this article with a new combina- torial 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 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, 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.

Author: ** Matthieu Josuat-Vergès - Jang Soo Kim - **

Source: https://hal.archives-ouvertes.fr/