A note on the number of representations of a positive integer as a sum of generalized polygonal numbers (Q6546692)

\textit{S. K. Jha} obtained identities connecting certain sums over the divisors of a positive integer \(n\) to the number of representations of \(n\) as a sum of squares [Rocky Mt. J. Math. 51, No. 2, 581--583 (2021; Zbl 1479.11021)] or a sum of triangular numbers [``An identity involving number of representations of \(n\) as a sum of \(r\) triangular numbers'', arXiv:2011.11038]. Those results are generalized here to give analogous identities for sums of \(s\)-gonal numbers for any integer \(s\geq 3\). The proofs involve Ramanujan's theta function and partial Bell polynomials. The results of Jha are obtained as corollaries.