For the last seminar of the year, we are proud to host Rahul Kumar. He has recently defended his thesis in IIT, Gandhinagar under the direction of Atul Dixit, one of the co-organizers of this seminar. The announcement appears below.
Talk Announcement
Title: A generalized modified Bessel function and explicit transformations of certain Lambert series
Speaker: Rahul Kumar (IIT, Gandhinagar)
When: December 17, 2020 - 3:55 PM - 5:00 PM (IST)
Where: Google Meet; Please write to sfandnt@gmail.com if you want a link.
Tea or Coffee: Please bring your own.
ABSTRACT:
An exact transformation, which we call a master identity, is obtained for the series $\sum_{n=1}^{\infty}\sigma_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. As corollaries when $a$ is an odd integer, we derive the well-known transformations of the Eisenstein series on $\text{SL}_{2}\left(\mathbb{Z}\right)$, that of the Dedekind eta function as well as Ramanujan's famous formula for $\zeta(2m+1)$. 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 $\zeta(2m+1)$. 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 $K_{\nu}(z)$. 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 $r_k(n)$. This is joint work with Atul Dixit and Aashita Kesarwani.
