Mathematics – Combinatorics
Scientific paper
2011-01-28
Mathematics
Combinatorics
29 pages
Scientific paper
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.
Josuat-Vergès Matthieu
Kim Jang Soo
No associations
LandOfFree
Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-552766