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

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