Relations between squares and triangles
type: Item
instance of: scholarly article
title: Relations between squares and triangles
zbMATH Open document ID: 0991.11014
DOI: 10.1016/S0012-365X(01)00344-2
author: Shaun Cooper
author: Michael D. Hirschhorn
author: Pierre Barrucand
published in: Discrete Mathematics
publication date: 2002-05-28T00:00:00Z
review text: The authors derive the interesting identities \[ r_k(8n+k)=2^{k-1} \left\{2+\binom{k}{4}\right\}t_k(n),\tag{1} \] where \(r_k(n)\) and \(t_k(n)\) are the number of representations of \(n\) as \(k\) squares and triangular numbers, respectively. The proof of the above uses the \(2\)-dissection of the function \[ \varphi(q)=\sum_{k=-\infty}^\infty q^{k^2} \] given by \[ \varphi(q)=\varphi(q^4)+2q\psi(q^8),\tag{2} \] where \[ \psi(q) = \sum_{k=0}^\infty q^{k(k+1)/2}. \] The authors first expand \(\varphi^k(q)\) using (2) and then extract terms congruent to \(k\pmod{8}\) in the expansion to derive (1).
reviewed by: Heng Huat Chan
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11A99
zbMATH DE Number: 1746263
zbMATH Keywords: sums of squares
zbMATH Keywords: sums of triangular numbers
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1746263