Relations between squares and triangles
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).
scientific article; zbMATH DE number 1746263