Moment sequences of Beta distribution

TL;DR

This paper explores the moment sequences of Beta distributions, linking them to well-known integer sequences like Catalan and Motzkin numbers, and uncovers new relationships through density ratio expansions.

Contribution

It identifies and connects Beta distribution moment sequences with famous integer sequences, revealing new mathematical relationships.

Findings

Identified around 20 Beta moment sequences as known integer sequences.

Established connections between Beta moments and Catalan, Motzkin, Riordan numbers.

Derived new relationships between these sequences using density ratio expansion.

Abstract

We recall some basic properties of the Beta distribution and some of its modifications. We identified around $20$ of the moment sequences of Beta distributions as important integer sequences in the OEIS base of integer sequences. Among those identified are Catalan, Riordan, Motzkin, or 'super ballot numbers'. By applying a method of expansion of the ratio of densities of involved distributions we are able to obtain some known and many unknown relationships between e.g. Catalan numbers and other moment sequences of the Beta distributions.