A generalized modified Bessel function and explicit transformations of certain Lambert series

Abstract

An exact transformation, which we call a \emph{master identity}, is obtained for the series ??n=1?a(n)e?ny for a?C and Re(y)>0. As corollaries when a is an odd integer, we derive the well-known transformations of the Eisenstein series on $\textup{SL}_{2}\left(\mathbb{Z}\right)$, that of the Dedekind eta function as well as Ramanujan's famous formula for ?(2m+1). Corresponding new transformations when a is a non-zero even integer are also obtained as special cases of the master identity. These include a novel companion to Ramanujan's formula for ?(2m+1). Although not modular, it is surprising that such explicit transformations exist. The Wigert-Bellman identity arising from the a=0 case of the master identity is derived too. The latter identity itself is derived using Guinand's version of the Vorono\"{\dotlessi} summation formula and an integral evaluation of N.~S.~Koshliakov involving a generalization of the modified Bessel function K?(z). Koshliakov's integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new transformation involving the sums-of-squares function rk(n). Some results on functions self-reciprocal in the Watson kernel are also obtained.