Self-reciprocal functions and double Mordell integrals

TL;DR

This paper explores self-reciprocal functions to evaluate Mordell type integrals, deriving explicit formulas and eigenfunctions for double cosine Fourier transforms, advancing the analytical tools for these integrals.

Contribution

It introduces new eigenfunctions of the double cosine Fourier transform and provides closed-form evaluations and reduction formulas for Mordell type integrals.

Findings

Identified two eigenfunctions of the double cosine Fourier transform.

Derived closed-form evaluations of certain Mordell integrals.

Established a reduction formula linking double and one-dimensional Mordell integrals.

Abstract

The theory of self-reciprocal functions is applied to the study Mordell type integrals. We find two particular eigenfunctions of the double cosine Fourier transform and then use them to evaluate certain one- and two-dimensional Mordell type integrals in closed form. A reduction formula is given for a certain family of double Mordell integrals in terms of one-dimensional integrals.