Yet another way of calculating moments of the Kesten's distribution and its consequences for Catalan numbers and Catalan triangles

TL;DR

This paper introduces a novel method for calculating moments of Kesten's distribution, revealing new identities involving Catalan numbers, triangles, and related combinatorial sequences with applications to Lucas, Fibonacci, and Fine numbers.

Contribution

It presents a new approach to compute moments of Kesten's distribution and derives novel identities linking Catalan-related numbers and polynomials.

Findings

Derived new identities involving Catalan numbers and polynomials

Established relations between Kesten's distribution moments and classical combinatorial sequences

Connected moments of Kesten's distribution to Lucas, Fibonacci, and Fine numbers

Abstract

We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles, binomial coefficients. Finally, as applications of these identities we obtain various interesting relations between the aforementioned numbers, also concerning Lucas, Fibonacci and Fine numbers.