Refine search
Actions for selected content:
1 results
Cooper–Hirschhorn type identities for three or more squares. III
- Part of
-
- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 69 / Issue 4 / November 2026
- Published online by Cambridge University Press:
- 22 May 2026, pp. 1183-1204
-
- Article
-
- You have access
- HTML
- Export citation
-
Let
$r_s(n)$ denote the number of representations of 
$n$ as a sum of 
$s$ squares. Hurwitz established eleven identities expressing the generating function of 
$r_3(an+b)$ as a simple infinite product. Cooper and Hirschhorn (Discrete Math 274 (1-3):9–24, 2004) proved that for any 
$k\geq0$, the generating functions 
$\sum_{n=0}^\infty r_3\big(3^{2k}n\big)q^n$ and 
$\sum_{n=0}^\infty r_3\big(3^{2k+1}n\big)q^n$ can be written as linear combinations of two specified generalized eta-quotients. In this paper, we substantially extend these results to high dimensions. Specifically, we prove that for any 
$k\geq0$ and 
$3\leq s\leq 100$, the generating functions 
$\sum_{n=0}^\infty r_s\big(3^{2k+1}n\big)q^n$ and 
$\sum_{n=0}^\infty r_s\big(3^{2k+2}n\big)q^n$ can also be expressed as linear combinations of certain generalized eta-quotients. Motivated by these results, we conjecture that this phenomenon holds for 
$r_s(n)$ for all 
$s\geq3$, and further that 
$r_s(n)$ satisfies an infinite family of internal congruences modulo high powers of 
$3$.
