Three classical results on representations of a number
The number of representations of a positive integer in the forms \(x^2+3y^2,\) \(x^2+xy+y^2,\) with \(x,y\in \mathbb Z\) and \((x^2+x)/2+3(y^2+y)/2\) with \(x,y\in \mathbb {Z}^+\) and various corresponding formulae as classical results are well-known at present. However, there exist also different methods of investigating this problem. The expressions of the formulae for the representations show the close relations between them. The author proves one known formulae of such type, namely a functional analog with the corresponding \(q\)-series and demonstrates the connections between them in such a way.
scientific article; zbMATH DE number 1301082