Journal of Integer Sequences, Vol. 22 (2019), Article 19.1.8

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

David Stenlund
Mathematics and Statistics
Åbo Akademi University
FI-20500 Åbo

James G. Wan
Engineering Systems and Design
Singapore University of Technology and Design
8 Somapah Road, 487372
School of Mathematical and Physical Sciences
The University of Newcastle
University Drive
Callaghan NSW 2308


In this paper we discuss a class of double sums involving ratios of binomial coefficients. The sums are of the form

\begin{displaymath}\sum_{j=0}^{n} \sum_{i=0}^j \frac{\binom{f_1(n)}{i}}{\binom{f_2(n)}{j}}\,c^{i-j}, \end{displaymath}

where f1, f2 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.

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(Concerned with sequences A001008 A002805 A027611 A046825 A046826 A096617 A233470 A234600.)

Received September 25 2018; revised version received January 24 2019. Published in Journal of Integer Sequences, February 15 2019.

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