Matthieu Josuat-Vergès ; Jang-Soo Kim - Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity

dmtcs:2934 - Discrete Mathematics & Theoretical Computer Science, January 1, 2011, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) - https://doi.org/10.46298/dmtcs.2934
Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identityArticle

Authors: Matthieu Josuat-Vergès ORCID1; Jang-Soo Kim 2

  • 1 Fakultät für Mathematik [Wien]
  • 2 School of Mathematics

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.


Volume: DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)
Section: Proceedings
Published on: January 1, 2011
Imported on: January 31, 2017
Keywords: Jacobi's triple product identity,continued fractions,enumeration,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Compact enumeration formulas for generalized partitions; Funder: Austrian Science Fund (FWF); Code: Y 463

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