On generalized Stirling numbers

Abstract

Let (z|α)n=z(z−α)⋯(z−nα+α). A "Stirling type pair``\n{S1(n,k),S2(n,k)}={S(n,k;α,β,r), S(n,k;β,α,−r)}\nis defined by means of\n(t|α)n=∑k=0nS1(n,k)(t−r|β)k,(t|β)n=∑k=0nS2(n,k)(t+r|α)k. By specializing the parameters α, β and r, one can obtain the Stirling numbers and various generalizations of the Stirling numbers. These definitions are not new; for example, S1(n,k) and S2(n,k) were defined and studied in [L. C. Hsu and H. Q. Yu, Appl. Math. J. Chinese Univ. Ser. B 12 (1997), no. 2, 225–232; MR1460101], where two of the generating functions in the present paper are given. See also [L. C. Hsu and P. J.-S. Shiue, Adv. in Appl. Math. 20 (1998), no. 3, 366–384; MR1618435], where many properties of S1 and S2 are worked out. Evidently, the paper under review is mainly concerned with proving generating functions and asymptotic expansions.