New generalized Mehler-Fock transformations and applications to the resolvent equation

TL;DR

This paper introduces generalized Mehler-Fock transformations involving Legendre functions, providing explicit formulas for Green's functions and heat kernels in hyperbolic wedges, advancing solutions to related resolvent equations.

Contribution

It develops a new class of integral transforms based on Legendre functions and applies them to solve heat and resolvent equations in hyperbolic geometry.

Findings

Explicit formulas for Green's functions in hyperbolic wedges

New generalized Mehler-Fock transformations introduced

Applications to heat kernel calculations in hyperbolic geometry

Abstract

We investigate and solve a special class of integrals involving associated Legendre functions, which can be regarded as generalized Mehler-Fock transformations. Some of the integrals appear naturally when dealing with the heat or resolvent equation of a wedge in the hyperbolic plane. As an application, we derive explicit formulas for the Green's function and the heat kernel of a wedge.