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 ...

In this paper, we establish several properties of the unified generalized Stirling numbers of the first kind, and the Jacobi-Stirling numbers of the first kind, by means of the convolution principle of sequences. Obtained results include generalized Vandermonde convolution for the unified generalized Stirling numbers of the first kind, triangular recurrence relation for general Stirling-type nu...

and that the Bernoulli numbers of the second kind bn for n > 0 may be generated by x ln(1+x) = ∑ ∞ n=0 bnx . In combinatorics, the signed Stirling number of the first kind s(n, k) may be defined such that the number of permutations of n elements which contain exactly k permutation cycles is the nonnegative number |s(n, k)| = (−1)s(n, k). The Bernoulli numbers of the second kind bn are also call...

The notion of the negative q-binomial was recently introduced by Fu, Reiner, Stanton and Thiem. Mirroring the negative q-binomial, we show the classical q-Stirling numbers of the second kind can be expressed as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in q and 1 + q. We extend this enumerative result via a decomposition of the Stirli...

Lengyel introduced a sequence of numbers Zn, defined combinatorially, that satisfy a recurrence where the coefficients are Stirling numbers of the second kind. He proved some 2-adic properties of these numbers. In this paper, we give another recurrence for the sequence Zn, where the coefficients are Stirling numbers of the first kind. Using this formula, we give another proof of Lengyel’s lower...