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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 theclassical modular substitution given by $q^*=e^{-2\pi i/\tau}$ and $\iota_qx=e^{2\pi i\xi/{\tau}}$. The main goal of this Note is to study the "modularbehaviour" of the infinite product $(x;q)_\infty$, this means, to compare thefunction 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 toa "closed" analytic formula for $(x;q)_\infty$ by means of the dilogarithmcombined with a Laplace type integral that admits a divergent series as Taylorexpansion at $\log q=0$. Thus, we can obtain an expression linking$(x;q)_\infty$ to its modular transform $(\iota_qx;q^*)_\infty$ and whichcontains, in essence, the modular formulae known for Dedekind's eta function,Jacobi theta function and also for certain Lambert series. Among otherapplications, one can remark that our results allow to obtain a Ramanujan'sasymptotic formula about $(x;q)_\infty$ for $q\to 1$.

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