Zeros of generalized Rogers-Ramanujan series: asymptotic and combinatorial properties
type: Item
instance of: scholarly article
title: Zeros of generalized Rogers-Ramanujan series: asymptotic and combinatorial properties
zbMATH Open document ID: 1208.30007
author: Tim Huber
published in: Journal of Approximation Theory
publication date: 2010-06-17T00:00:00Z
review text: The author studies the Rogers-Ramanujan series \[ R(x,q):=\sum_{n=0}^{\infty}\frac{q^{n^2}x^n}{(q;q)_n},\quad (\alpha;q)_n:=\prod_{k=0}^{n-1}(1-\alpha q^k),\quad|q|<1. \] In his famous Lost Notebook, Ramanujan mentioned a Hadamard product expansion of the form \[ R(x,q)=\prod_{n=1}^{\infty}\left(1+\frac{xq^{2n-1}}{1-\sum_{i=1}^{\infty}y_iq^{ni}}\right) \] with explicit formulas for the first four coefficients \(\{y_i\}_{i=1}^4\) in terms of the function \[ \psi(q):=\sum_{n=0}^{\infty}q^{n(n+1)/2}=\frac{(q^2;q^2)_{\infty}}{(q;q^2)_{\infty}}. \] It is shown that for \(q\) tending to \(1\) the coefficients \(y_n\) tend to a rational multiple of \(\pi^{-1}\) for \(n\) odd, and to a rational multiple of \(\pi^{-2}\) for \(n\) even. This is a slightly different form of a conjecture given in [\textit{M. E. H. Ismail} and \textit{C. Zhang}, Adv. Math. 209, No. 1, 363--380 (2007; Zbl 1107.33018)]. It is also shown that the coefficients \(y_i\) are connected to a \(q\)-analogue of the tangent numbers \(T_n\) appearing in the Taylor expansion \[ \tan x=\sum_{n=0}^{\infty}\frac{T_{2n+1}x^{2n+1}}{(2n+1)!}. \] Finally, a connection with Jacobian elliptic functions leads to several recursion formulas for the calculation of series expansions for the zeros of \(R(x,q)\). This is motivated by the earlier mentioned work of Ismail and Zhang.
reviewed by: Roelof Koekoek
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 33D15
Mathematics Subject Classification ID: 33D45
zbMATH DE Number: 5722074
zbMATH Keywords: Rogers-Ramanujan series
zbMATH Keywords: zeros
zbMATH Keywords: asymptotic properties
zbMATH Keywords: elliptic integrals
MaRDI profile type: Publication
full work available at URL: j.jat.2009.10.001
OpenAlex ID: W2001645704
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DOI: 10.1016/J.JAT.2009.10.001
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description: scientific article; zbMATH DE number 5722074