Integrals derived from the doubling method

TL;DR

This paper explains the derivation of various global Rankin-Selberg integrals from a basic identity related to the doubling method, using Eisenstein series and root exchange, and introduces a new integral example.

Contribution

It provides a unified framework for deriving known integrals and introduces a new global integral using the doubling method and identities involving Eisenstein series.

Findings

Unified derivation of known Rankin-Selberg integrals

Introduction of a new global integral example

Clarification of the role of the doubling method in integral construction

Abstract

In this note, we use a basic identity, derived from the generalized doubling integrals of \cite{C-F-G-K1}, in order to explain the existence of various global Rankin-Selberg integrals for certain $L$-functions. To derive these global integrals, we use the identities relating Eisenstein series in \cite{G-S}, together with the process of exchanging roots. We concentrate on several well known examples, and explain how to obtain them from the basic identity. Using these ideas, we also show how to derive a new global integral.