Introduction. We shall consider functions of the form
where {r
i
} and {s
i
} are sets of positive integers. Such functions were studied by E. Grosswald in [2], who took {s
i
} to be pairwise relatively prime, and asked the following two questions:
(a) When is ƒ(t) a polynomial?
(b) When does ƒ(t) have positive coefficients?
These questions arise naturally from the work of Allday and Halperin, who show in [1] that under suitable circumstance ƒ(t) will be the Poincare polynomial of the orbit space of a certain Lie group action. Grosswald gives a complete answer to (a), but (b) is a much harder question, and a complete answer is provided only for the case m = 2. His treatment involves the representation of the coefficients of ƒ(t) by partition functions, and uses a classical description by Sylvester of the semigroup generated by {s
i
}.