In this paper, several explicit formulas for the number of representations of a positive integer by sums of mixed numbers are determined by employing theta function identities and the [Formula: see text]-parametrization of theta functions due to Alaca, Alaca and Williams. It is interesting that the formulas proved in this paper are linear combinations of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].