On twisted Koecher-Maass series and its integral Kernel

TL;DR

This paper develops the integral kernel for degree three twisted Koecher-Maass series, proves its analytic continuation and functional equations, and offers new representations and applications, including a generalization of Siegel's Lipschitz summation formula.

Contribution

It introduces a new integral kernel for twisted Koecher-Maass series of degree three, with analytic continuation, functional equations, and alternative representations, advancing understanding of these series.

Findings

Established the integral kernel for twisted Koecher-Maass series of degree three.

Proved the kernel admits an analytic continuation and derived its functional equations.

Provided a new proof of the analytic properties of the twisted Koecher-Maass series.

Abstract

We establish the integral kernel associated with the Koecher-Maass series of degree three twisted by an Eisenstein series. We prove that such a kernel admits an analytic continuation and determine its functional equations. We find a second representation of this kernel using Poincar\'e series. As an application, we give another proof of the analytic properties for the twisted Koecher-Maass series. Furthermore, we generalize a result due to Siegel related to the Lipschitz summation formula on the Siegel space of degree three.