Arithmetic properties arising from Ramanujan's theta functions (Q523695)

The generating functions for the number of representations of \(n\in\mathbb{N}\) as \(\triangle_1+2k\triangle_2\) and \(\triangle+k\square\) are well-formulated in terms of Ramanujan's theta functions \(\varphi(q)\) and \(\psi(q)\). The present work looks at the representation \(\triangle+k\square\) when \(n\) satisfies congruence properties viz. \(n\equiv 1\pmod{3}\), \(n\equiv 3\pmod{5}\), \(n\equiv 6\pmod{7}\) for \(k=1,2,3\) respectively, and at the representation \(\triangle_1+2k\triangle_2\) for all numbers of the form \((2k+1)n\) for fixed \(k\). Along with the relation \(\varphi(q)=\varphi(q^4)+2q\psi(q^8)\), the main tool used in the derivations is the fact that the solution of \(x^2+2ky^2\equiv0\pmod{2k+1}\) is \(x\equiv\pm y\pmod{2k+1}\). As consequences, congruences modulo powers of \(2\) for the number of representations of \(n\in\mathbb{N}\) as \(\triangle_1+4\triangle_2\) and \(\triangle+k\square\) are also obtained.