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The Ramanujan Journal

, Volume 46, Issue 1, pp 269–305 | Cite as

A modular-type formula for \((x;q)_\infty \)

  • Changgui Zhang
Article
  • 140 Downloads

Abstract

Let \(q=\text {e}^{2\pi i\tau }, \mathfrak {I}\tau >0\), \(x=\text {e}^{2\pi i{z}}\), \({z}\in \mathbb {C}\), and \((x;q)_\infty =\prod _{n\ge 0}(1-xq^n)\). Let \((q,x)\mapsto ({q_1},{x_1})\) be the classical modular substitution given by the relations \({q_1}=\text {e}^{-2\pi i/\tau }\) and \({x_1}=\text {e}^{2\pi i{z}/{\tau }}\). The main goal of this paper is to give a modular-type representation for the infinite product \((x;q)_\infty \), this means, to compare the function defined by \((x;q)_\infty \) with that given by \(({x_1};{q_1})_\infty \). Inspired by the work (Stieltjes in Collected Papers, Springer, New York, 1993) of Stieltjes on semi-convergent series, we are led to a “closed” analytic formula for the ratio \((x;q)_\infty /({x_1};{q_1})_\infty \) by means of the dilogarithm combined with a Laplace type integral, which admits a divergent series as Taylor expansion at \(\log q=0\). Thus, the function \((x;q)_\infty \) is linked with its modular transform \(({x_1};{q_1})_\infty \) in such an explicit manner that one can directly find the modular formulae known for Dedekind’s Eta function, Jacobi Theta function, and also for certain Lambert series. Moreover, one can remark that our results allow Ramanujan’s formula (Berndt in Ramanujan’s notebooks, Springer, New York, 1994, Entry 6’, p. 268) (see also Ramanujan in Notebook 2, Tata Institute of Fundamental Research, Bombay, 1957, p. 284) to be completed as a convergent expression for the infinite product \((x;q)_\infty \).

Keywords

Modular elliptic functions Jacobi \(\theta \)-function Dedekind \(\eta \)-function Lambert series q-series 

Mathematics Subject Classification

11F20 11F27 30D05 33E05 05A30 

Notes

Acknowledgements

The author would like to express thanks to Lucia Di Vizio, Anne Duval, Jean-Pierre Ramis, and Jacques Sauloy for their numerous valuable suggestions and remarks, and express thanks to the referee for the formulae in (1.3) and (1.4).

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Laboratoire P. Painlevé CNRS UMR 8524, UFR de MathématiquesUniversité des Sciences et Technologies de LilleVilleneuve d’Ascq CedexFrance

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