On the Modular Behaviour of the Infinite Product $(1-x)(1-xq)(1-xq^2)(1-xq^3)...$

Abstract

Let $q=e^{2\pi i\tau}$, $\Im\tau>0$, $x=e^{2\pi i\xi}\in\CC$ and $(x;q)_\infty=\prod_{n\ge 0}(1-xq^n)$. Let $(q,x)\mapsto(q^*,\iota_q x)$ be the classical modular substitution given by $q^*=e^{-2\pi i/\tau}$ and $\iota_q x=e^{2\pi i\xi/{\tau}}$. The main goal of this Note is to study the "modular behaviour" of the infinite product $(x;q)_\infty$, this means, to compare the function defined by $(x;q)_\infty$ with that given by $(\iota_q x;q^*)_\infty$. Inspired by the work of Stieltjes on some semi-convergent series, we are led to a "closed" analytic formula for $(x;q)_\infty$ by means of the dilogarithm combined with a Laplace type integral that admits a divergent series as Taylor expansion at $\log q=0$. Thus, we can obtain an expression linking $(x;q)_\infty$ to its modular transform $(\iota_qx;q^*)_\infty$ and which contains, in essence, the modular formulae known for Dedekind's eta function, Jacobi theta function and also for certain Lambert series. Among other applications, one can remark that our results allow to obtain a Ramanujan's asymptotic formula about $(x;q)_\infty$ for $q\to 1$.