Asymptotic expansions of Lambert series and related q-series

作者：Banerjee Shubho^*; Wilkerson Blake

来源：International Journal of Number Theory, 2017, 13(8): 2097-2113.

DOI：10.1142/S1793042117501135

• Lambert
• Eisenstein
• q-series
• q-gamma
• q-Pochhammer
• Jacobi theta

We study the Lambert series L-q(s, x) = Sigma(infinity)(k=1) k(s)q(kx)/(1 - q(k)), for all s is an element of C. We obtain the complete asymptotic expansion of L-q(s, x) near q = 1. Our analysis of the Lambert series yields the asymptotic forms for several related q-series: the q-gamma and q-polygamma functions, the q-Pochhammer symbol and the Jacobi theta functions. Some typical results include Gamma(2) (1/4)Gamma(2) (3/4) approximate to 2(13/32)pi/log 2 and theta(4)(0, e(-1/pi)) approximate to 2 pi e(-pi 3/4), with relative errors of order 10(-25) and 10(-27) respectively.

• 出版日期2017-9