Unclosed asymptotic expansions and mock theta functions |
| |
Authors: | Richard J Mcintosh |
| |
Institution: | (1) Department of Mathematics and Statistics, University of Regina, 10-3737 Wascana parkway, Regina, SK, S4S 0A2, Canada |
| |
Abstract: | In his last letter to Hardy, Ramanujan defined 17 functions f(q), (|q|<1), which he called mock theta functions. Each f(q) has infinitely many exponential singularities at roots of unity, and under radial approach to every such singularity, f(q) has an asymptotic approximation consisting of a finite number of terms with closed exponential factors, plus an error term
O(1). We give an example of a q-series in Eulerian form having an approximation with an unclosed exponential factor. Complete asymptotic expansions as q→1 of some shifted q-factorials are given in terms of polylogarithms and Bernoulli polynomials.
Supported by the Natural Sciences and Engineering Research Council of Canada. |
| |
Keywords: | Mock theta function Asymptotics Polylogarithm Bernoulli polynomial |
本文献已被 SpringerLink 等数据库收录! |
|