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Lambert series and q-functions near q=1 (1602.01085v1)
Published 2 Feb 2016 in math.NT
Abstract: We study the Lambert series $\mathscr{L}q(s,x) = \sum{k=1}\infty ks q{k x}/(1-qk)$, for all $s \in \mathbb{C}$. We obtain the complete asymptotic expansion of $\mathscr{L}_q(s,x)$ near $q=1$. Our analysis of the Lambert series yields the asymptotic forms for several related q-functions: the q-gamma and q-polygamma functions, the q-Pochhammer symbol, and, in closed form, the Jacobi theta functions. Some typical results include $\Gamma_2(\frac{1}{4}) \Gamma_2(\frac{3}{4}) \simeq \frac{2{13/32} \pi}{\log 2}$ and $\vartheta_4 (0,e{-1/\pi}) \simeq 2 \pi e{-\pi3!/4}$, with relative errors of order $10{-25}$ and $10{-27}$ respectively.