A new Andrews--Crandall-type identity and the number of integer solutions to $x^2+2y^2+2z^2=n$

TL;DR

This paper introduces a novel identity inspired by Kronecker's work, enabling the precise counting of integer solutions to the quadratic form x^2+2y^2+2z^2=n, advancing understanding in number theory.

Contribution

The paper presents a new Andrews--Crandall-type identity derived from a higher-dimensional analog of Kronecker's identity, specifically for counting solutions to a quadratic form.

Findings

Derived a new identity for counting solutions to x^2+2y^2+2z^2=n

Provided explicit formulas for the number of solutions

Enhanced methods for analyzing quadratic forms in number theory

Abstract

Using a higher-dimensional analog of an identity known to Kronecker, we discover a new Andrews--Crandall-type identity and use it to count the number of integer solutions to $x^2+2y^2+2z^2=n$.