A unified approach to the integrals of Mellin--Barnes--Hecke type

TL;DR

This paper introduces a unified method using distribution theory and Fourier transforms to evaluate Mellin--Barnes type integrals, revealing new connections with Bessel functions relevant to number theory.

Contribution

It develops a comprehensive approach employing distribution pull-backs to evaluate Mellin--Barnes integrals, unifying various special cases involving Bessel functions.

Findings

Derived integrals of Hecke and Sonine involving Bessel functions

Unified framework simplifies evaluation of Mellin--Barnes integrals

Applications in analytic and algebraic number theory

Abstract

In this paper we provide a unified approach to a family of integrals of Mellin--Barnes type using distribution theory and Fourier transforms. Interesting features arise in many of the cases which call for the application of pull-backs of distributions via smooth submersive maps defined by H\"ormander. We derive by this method the integrals of Hecke and Sonine relating to various types of Bessel functions which have found applications in analytic and algebraic number theory.