Some Double Sums Involving Ratios of Binomial Coefficients Arising From Urn Models

TL;DR

This paper analyzes a class of double sums involving ratios of binomial coefficients related to urn models, simplifying them using hypergeometric functions and harmonic numbers to aid probabilistic analysis.

Contribution

It introduces a method to evaluate complex double sums involving binomial ratios using hypergeometric functions and harmonic numbers, applicable to urn models.

Findings

Derived simplified expressions for double sums involving binomial ratios.

Connected sums to hypergeometric functions and harmonic numbers.

Provided tools for probabilistic analysis of urn models.

Abstract

In this paper we discuss a class of double sums involving ratios of binomial coefficients. The sums are of the form \[ \sum_{j=0}^{n} \sum_{i=0}^j \frac{\binom{f_1(n)}{i}}{\binom{f_2(n)}{j}}\,c^{i-j}, \] where $f_1, f_2$ are functions of $n$. Such sums appear in the analyses of the Mabinogion urn and the Ehrenfest urn in probability. Using hypergeometric functions, we are able to simplify these sums, and in some cases express them in terms of the harmonic numbers.