Recurrence relations for polynomials obtained by arithmetic functions

Families of polynomials associated to arithmetic functions [Formula: see text] are studied. The case [Formula: see text], the divisor sum, dictates the non-vanishing of the Fourier coefficients of powers of the Dedekind eta function. The polynomials [Formula: see text] are defined by [Formula: see text]-term recurrence relations. For the case that [Formula: see text] is a polynomial of degree [Formula: see text], we prove that at most a [Formula: see text] term recurrence relation is needed. For the special case [Formula: see text], we obtain explicit formulas and results.