arXiv subjects

James G. Wan

Publications and source records attributed to James G. Wan.

3 recordsLinked to original sources

arXiv2018

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.

math.CO↗

arXiv2016

Holonomic alchemy and series for $1/π$

We adopt the "translation" as well as other techniques to express several identities conjectured by Z.-W. Sun in arXiv:1102.5649v47 by means of known formulas for $1/π$ involving Domb and other Apéry-like sequences.

math.NT↗

arXiv2013

Series for $1/π$ Using Legendre's Relation

We present a new method for producing series for $1/π$ and other constants using Legendre's relation, starting from a generation function that can be factorised into two elliptic $K$'s; this way we avoid much of modular theory or creative telescoping. Many of our series involve special values of Legendre polynomials; their relationship to the more traditional Ramanujan series is discussed.

math.NT↗