On a product of three theta functions and the number of representations of integers as mixed ternary sums involving squares, triangular, pentagonal and octagonal numbers

TL;DR

This paper develops formulas to express products of three theta functions as linear combinations, enabling new results on representing integers as sums involving squares, triangular, pentagonal, and octagonal numbers.

Contribution

It introduces a general formula for products of three theta functions and applies it to derive new theorems on integer representations with mixed figurate numbers.

Findings

Derived a general formula for the product of three theta functions

Established a formula for the product of two theta functions

Proved new theorems on mixed sums involving squares and figurate numbers

Abstract

In this paper, we derive a general formula to express the product of three theta functions as a linear combination of other products of three theta functions. Moreover, we use the main formula to deduce a general formula for the product of two theta functions. Furthermore, as applications, we extract several theorems in the theory of representation of integers as mixed ternary sums involving squares, triangular numbers, generalized pentagonal numbers and generalized octagonal numbers