On a Kind of Generalized Arithmetic-geometric Progression

whered and h are real or complex numbers, and/? a positive integer. It is always assumed that a * Q and a * 1. It is known that the sum Sa^p(n) = J?k=llak has been investigated with different methods by de Bruyn [1] and Gauthier [6]. De Bruyn developed some explicit formulas by using certain determinant expressions derived from Cramer's rule, and Gauthier made repeated use of the differential operator D = x(d I dx) to express Sa^p(n) as a times a polynomial of degree/? in ny plus an w-independent term in which the coefficients are determined recursively. In this note we shall express the general sum Sahp(ri) explicitly in terms of the degenerate Stirling numbers due to Carlitz [2]. In particular, an explicit formula for Sa0p(n) will be given via Stirling numbers of the second kind. Finally, as other applications of our Lemma 1, some combinatorial sums involving generalized factorials will be presented.