Andrew Sills, Georgia Southern University

Abstract

In The Theory of Partitions, p. 98, Ex. 2, George Andrews points out that any power series with constant term 1 has a unique factorization in the form 1 + r(1)*q + r(2)*q^2 + r(3)*q^3 + . . . = (1-q)^{-a_1} * (1-q^2)^{-a_2} * (1-q^3)^{-a^3} * . . . . He then suggests an algorithm to calculate the r(n) given the a_i . In his qseries.m Maple package, Frank Garvan programmed the inverse algorithm, i.e. given the r(n), find the a_i. Shashank Kanade and Matthew Russell used this algorithm extensively in their discovery of many new Rogers--Ramanujan type identities, where the a_i form a discrete periodic function with respect to a fixed modulus. In 1954, G. Meinardus published an asymptotic formula for the r(n) in terms of the a_i. Recently, I found an exact formula for the r(n) in terms of the a_i and vice versa, which I will share after presenting some background material. This work is part of a larger ongoing project joint with Robert Schneider and Hunter Waldron of Michigan Tech.

Link to video: https://vimeo.com/1027498379?share=copy

Presented Via Zoom: https://rutgers.zoom.us/j/91865817691

Password: 6564120420

For further information see: https://sites.math.rutgers.edu/~zeilberg/expmath/