A modified approach via differential operator is given to derive a generalization of Stirling numbers of the first kind. This approach combines the two techniques given by Cakic [3] and Blasiak [2]. Some new combinatorial identities and many relations between different types of Stirling numbers are found. Furthermore, some interesting special cases of the generalized Stirling numbers of the fir...

Let s(n,m) denote the (unsigned) Stirling numbers of the first kind: s(n,m) := [w] (w(w + 1) · · · (w + n− 1)) (1 ≤ m ≤ n, n ≥ 1). Many different asymptotic expressions for s(n,m), as n→∞, have been proposed in the literature due to their wide applications, cf. Temme [8] for a brief survey of known results together with a uniform asymptotic expansion valid for all m, 1 ≤ m ≤ n. Recently, Wilf [...

In this paper we extend the definition of Stirling numbers of the first kind by way of a special multiset. This results in a family of number triangles for which we show how to obtain ordinary generating functions for the rows and exponential generating functions for the columns. The latter are derived via a recursive process. We also indicate how to obtain formulas, in terms of factorials, gen...

It is by now well known that the parities of the binomial coefficients show a fractal-like appearance when plotted in the x-y plane. Similarly, if f(n,k) is some counting sequence and/? is a prime, we can plot an asterisk at (//,&) iff(n,k) # 0 (mod/?), and a blank otherwise, to get other complex, and often interesting, patterns. For the ordinary and Gaussian binomial coefficients and for the S...