On Log-concavity of a Class of Generalized Stirling Numbers

Abstract

This paper considers the generalized Stirling numbers of the first and second kinds. First, we show that the sequences of the above generalized Stirling numbers are both log-concave under some mild conditions. Then, we show that some polynomials related to the above generalized Stirling numbers are $q$-log-concave or $q$-log-convex under suitable conditions. We further discuss the log-convexity of some linear transformations related to generalized Stirling numbers of the first kind.

Some applications of the chromatic polynomials

“…i for all i > 0, and, it is called a Pólya-frequency sequence (or a PF sequence) if all minors of the matrix A = (u i−j ) i,j≥0 have nonnegative determinants (where u k = 0 if k < 0), for more information see [5]. A sequence of real polynomials (P n (q), n ≥ 0) is called q-log-convex if the polynomial P n (q) 2 − P n−1 (q)P n+1 (q) has nonnegative coefficients for all n ≥ 1, see [12,13,19]. Some known results on such sequences are given as follows.…”

The (r1,...,rp)-Bell polynomials

“…, the greatest maximizing index of n k rp and we give an approximation of n+|rp| m+rp rp when n tends to infinity. The case p = 1 was studied by Mező [9] and other study is given by Zhao [15] on a large class of Stirling numbers.…”