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Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity | | Matthieu Josuat-Vergès
; Jang Soo Kim
; | | Date: |
28 Jan 2011 | | Abstract: | Touchard-Riordan-like formulas are some expressions appearing in enumeration
problems and as moments of orthogonal polynomials. We begin this article with a
new combinatorial 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 identity 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 identity, 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. | | Source: | arXiv, 1101.5608 | | Services: | Forum | Review | PDF | Favorites | |
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