Multifold Convolutions, Generating Functions and 1d Random Walks

TL;DR

This paper explores the properties and asymptotics of multifold convolutions of combinatorial sequences like Catalan numbers and binomial coefficients, linking them to random walks and using complex analysis for asymptotic calculations.

Contribution

It introduces new asymptotic formulas for multifold convolutions of combinatorial sequences and applies complex analysis techniques to study their large deviations.

Findings

Explicit formulas for convolutions of Catalan numbers and binomial coefficients.

Asymptotic behavior of convolutions for large n.

Application of complex analysis to large deviation problems.

Abstract

We consider multifold convolutions of a combinatorial sequence $(a_n)_{n=0}^{\infty}$: namely, for each $k \in \N$ the $k$-fold convolution is $\mathcal{M}^{(k)}_n(\boldsymbol{a}) = \sum_{j_1+\dots+j_k=n} a_{j_1} \cdots a_{j_k}$. Let $C_n$ be the Catalan numbers, and let $B_n$ be the central binomial coefficients. Then for random Dyck paths or simple random walk bridges, the multifold convolutions give moments of returns to the origin, using the stars-and-bars problem. There are well-known explicit formulas for the multifold convolutions of $C_n$ and $B_n$. But even for combinatorial sequences $B_n^2$ and $B_n^3$, one may determine asymptotics of multifold convolutions for large $n$. We also discuss large deviations: In a second part of the paper we consider an elementary version of the circle method for calculating asymptotics using complex analysis.