Geodesic
Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (2011).

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We give a combinatorial proof of a Touchard-Riordan-like formula discovered by the first author. 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. 
@article{DMTCS_2011_special_260_a47,
     author = {Josuat-Verg\`es, Matthieu and Kim, Jang-Soo},
     title = {Touchard-Riordan formulas, {T-fractions,} and {Jacobi's} triple product identity},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)},
     year = {2011},
     doi = {10.46298/dmtcs.2934},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2934/}
}
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Josuat-Vergès, Matthieu; Kim, Jang-Soo. Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (2011). doi : 10.46298/dmtcs.2934. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2934/

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