A Generalization of Stirling Numbers

are called a weighted Stirling pair if f(g(x)) = g(f(x)) = x, i.e.,/and g are reciprocal. When W(x) = l, B^in.k) and B2(n,k) reduce to a Stirling type pair whose properties are exhibited in [7]. In this paper, we shall present a weighted Stirling pair that includes some previous generalizations of Stirling numbers as particular cases. Some related combinatorial and arithmetic properties are also discussed. 2. A WEIGHTED STIRLING PAIR Let t, a, P be given complex numbers with a-(1*0. Let f(x) = [(1 + ax)P' -1]//?, g(x) [(\+px)-l]la, and W(x) = (l + ax). Then, in accordance with (1), by noting that f(x) and g(x) are reciprocal, we have a weighted Stirling pair, denoted by {S(n, k, a, /?; t\ S(n, k, /?, a, -1)} = {£> , k), B2(n, k)}. We call it an (a, /?; /) [resp. a (ft, a; -t)] pair for short. Moreover, one of the parameters a or p may be zero by considering the limit process. For instance, a (1,0; 0) [resp. a (0,1; 0)] pair is just Stirling numbers of the first and second kinds. Note that from the definition of an (a, /?; t) pair and the first equation in (1), we may obtain the double generating function of S(n, k, a, /?; t) as