Banerjee, SoumyarupSoumyarupBanerjeeDixit, AtulAtulDixitGupta, ShivajeeShivajeeGupta2025-08-282025-08-282022-05-01http://arxiv.org/abs/2205.11351http://repository.iitgn.ac.in/handle/IITG2025/20110An explicit transformation for the series ?n=1?log(n)eny?1, Re(y)>0, which takes y to 1/y, is obtained for the first time. This series transforms into a series containing ?1(z), the derivative of Deninger's function R(z). In the course of obtaining the transformation, new important properties of ?1(z) are derived, as is a new representation for the second derivative of the two-variable Mittag-Leffler function E2,b(z) evaluated at b=1. Our transformation readily gives the complete asymptotic expansion of ?n=1?log(n)eny-1 as y?0. An application of the latter is that it gives the asymptotic expansion of ??0?(12-it)??(12+it)e-?tdt as ??0.en-USDeninger's functionAsymptotic expansionMittag-Leffler functionLambert seriesLogarithme-Printe-Print123456789/555