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- W1541514872 abstract "Let $q=e^{2pi itau}$, $Imtau>0$, $x=e^{2pi ixi}inCC$ and $(x;q)_infty=prod_{nge 0}(1-xq^n)$. Let $(q,x)mapsto(q^*,iota_q x)$ be the classical substitution given by $q^*=e^{-2pi i/tau}$ and $iota_q x=e^{2pi ixi/{tau}}$. 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 transform $(iota_qx;q^*)_infty$ and which contains, in essence, the 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 $qto 1$." @default.
- W1541514872 created "2016-06-24" @default.
- W1541514872 creator A5032525120 @default.
- W1541514872 date "2011-12-21" @default.
- W1541514872 modified "2023-09-27" @default.
- W1541514872 title "On the Modular Behaviour of the Infinite Product $(1-x)(1-xq)(1-xq^2)(1-xq^3)...$" @default.
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