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

We give a combinatorial proof of a Touchard-Riordan-like formula discovered by Josuat-Vergès. As a consequence we find a connection between his formula and Jacobi's triple product identity. We then give a combinatorial analog of Jacobi's triple product identity by showing that a finite sum can be interpreted as a generating function of weighted Schröder paths, so that the triple product identity is recovered by taking the limit. This can be stated in terms of some continued fractions called T-fractions, whose important property is the fact that they satisfy some functional equation. We show that this result permits to explain and generalize some Touchard-Riordan-like formulas appearing in enumerative problems. This is joint wort with Matthieu Josuat-Vergès.