377 research outputs found
Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity
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.Comment: 29 page
Crossings, Motzkin paths and Moments
Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain
-analogues of Laguerre and Charlier polynomials. The moments of these
orthogonal polynomials have combinatorial models in terms of crossings in
permutations and set partitions. The aim of this article is to prove simple
formulas for the moments of the -Laguerre and the -Charlier polynomials,
in the style of the Touchard-Riordan formula (which gives the moments of some
-Hermite polynomials, and also the distribution of crossings in matchings).
Our method mainly consists in the enumeration of weighted Motzkin paths, which
are naturally associated with the moments. Some steps are bijective, in
particular we describe a decomposition of paths which generalises a previous
construction of Penaud for the case of the Touchard-Riordan formula. There are
also some non-bijective steps using basic hypergeometric series, and continued
fractions or, alternatively, functional equations.Comment: 21 page
Approximations for the Moments of Nonstationary and State Dependent Birth-Death Queues
In this paper we propose a new method for approximating the nonstationary
moment dynamics of one dimensional Markovian birth-death processes. By
expanding the transition probabilities of the Markov process in terms of
Poisson-Charlier polynomials, we are able to estimate any moment of the Markov
process even though the system of moment equations may not be closed. Using new
weighted discrete Sobolev spaces, we derive explicit error bounds of the
transition probabilities and new weak a priori estimates for approximating the
moments of the Markov processs using a truncated form of the expansion. Using
our error bounds and estimates, we are able to show that our approximations
converge to the true stochastic process as we add more terms to the expansion
and give explicit bounds on the truncation error. As a result, we are the first
paper in the queueing literature to provide error bounds and estimates on the
performance of a moment closure approximation. Lastly, we perform several
numerical experiments for some important models in the queueing theory
literature and show that our expansion techniques are accurate at estimating
the moment dynamics of these Markov process with only a few terms of the
expansion
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