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

Matthieu Josuat-Vergès (1) , Jang Soo Kim (2)

1 Fakultät für Mathematik [Wien]
2 UMN-MATH - School of Mathematics

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.

Domaines: Combinatoire [math.CO]

Matthieu Josuat-Vergès, Jang Soo Kim. Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity. Ramanujan Journal, 2013, 30 (3), pp.341--378. ⟨10.1007/s11139-012-9403-9⟩. ⟨hal-00967539⟩