A Note on the Number of Representations of $n$ as a Sum of Generalized Polygonal Numbers

TL;DR

This paper generalizes recent divisor sum identities to relate the number of representations of an integer as a sum of s-gonal numbers for any s ≥ 3, extending previous specific cases involving squares and triangular numbers.

Contribution

It introduces a generalized formula connecting divisor sums to s-gonal number representations for all integers s ≥ 3, broadening prior specific results.

Findings

Established a generalized relation for s-gonal numbers

Extended identities from squares and triangular numbers to all s-gonal numbers

Provides a unified framework for divisor sum and polygonal number representations

Abstract

Recently, Jha (arXiv:2007.04243, arXiv:2011.11038) has found identities that connect certain sums over the divisors of $n$ to the number of representations of $n$ as a sum of squares and triangular numbers. In this note, we state a generalized result that gives such relations for $s$-gonal numbers for any integer $s\geq3$.