NOTE ON STIRLING POLYNOMIALS

A Note on Stirling Series

We study sums S = S(d, n, k) = ∑ j≥1 [ d] jk( j )j! with d ∈ N = {1, 2, . . . } and n, k ∈ N0 = {0, 1, 2, . . . } and relate them to (finite) multiple zeta functions. As a byproduct of our results we obtain asymptotic expansions of ζ(d + 1) −H n as n tends to infinity. Furthermore, we relate sums S to Nielsen’s polylogarithm.

On Generalized Stirling Numbers and Polynomials

In this paper we prove that some results concerned the generalized Stirling numbers are the consequence of the results of Toscano and Chak. The new explicit expressions for generalized Stirling numbers are also given.

Applications of Chromatic Polynomials Involving Stirling Numbers

The Stirling numbers of the second kind, denoted S(n, k), are the number of ways to partition n distinct objects into k nonempty subsets. We use the notation [n] = {1, 2,. . ., n} and sometimes refer to the subsets as blocks. The initial conditions are defined as: S(0, 0) = 1, S(n, 0) = 0, for n ≥ 1, and S(n, k) = 0 for k > n. We also have S(n, 2) = 2 n−1 − 1 and S(n, n − 1) = n 2. The numbers ...

Note on Certain Orthogonal Polynomials

A Note on Boole Polynomials

Boole polynomials play an important role in the area of number theory, algebra and umbral calculus. In this paper, we investigate some properties of Boole polynomials and consider Witt-type formulas for the Boole numbers and polynomials. Finally, we derive some new identities of those poly-nomials from the Witt-type formulas which are related to Euler polynomials.