On the irrationality of Cantor series

In this paper we study the (ir)rationality of sumswhere P (x) and Q(x) are polynomials with rational coefficients such that the sum is well-defined and Q assumes integer values at integer points.In Section 2 we derive criteria corresponding with Oppenheim's irrationality criteria for Cantor serieswith bn an → 0 as n → ∞.In Section 3 we present some sufficient conditions for irrationality.In Section 4 we deal with such conditions for some closely related sums.