Abstract In this paper, we discuss the properties of associated Stirling numbers. By means of the method of coefficients, we establish a series of identities involving associated Stirling numbers, Bernoulli numbers, harmonic numbers, and the Cauchy numbers of the first kind. In addition, we give the asymptotic expansion of certain sums involving 2-associated Stirling numbers of the second kind ...

n m n m Z ^ . W" = 5>(w, jy^Fip, k)(k)j = Zj\S(m, j)(n, j). k=o y=o k=o j=o Notice that the special case for m = 0 is also true. Hence, (2) holds for every m > 0. • Remark Sometimes in applications of the rule function F(n, k) may involve some independent parameters. Moreover, for the particular case in which F(n, k) > 0, so that (j)(n, 0) > 0, the lefthand side of (2) divided by (j)(n, 0) m...

Stirling numbers of the second kind and Bell numbers for graphs were defined by Duncan and Peele in 2009. In a previous paper, one of us, jointly with Nyul, extended the known results for these special numbers by giving new identities, and provided a list of explicit expressions for Stirling numbers of the second kind and Bell numbers for particular graphs. In this work we introduce q-Stirling ...

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...

In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q; r; w)Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q...

The Stirling number of the second kind {k} counts the number of ways to partition a set of n labeled balls into k non-empty unlabeled cells. We extend this problem and give a new statement of the r-Stirling numbers of the second kind and r-Bell numbers. We also introduce the r-mixed Stirling number of the second kind and r-mixed Bell numbers. As an application of our results we obtain a formula...

These theories introduce basic concepts and proofs about discrete summation: shifts, formal summation, falling factorials and stirling numbers. As proof of concept, a simple summation conversion is provided. 1 Stirling numbers of first and second kind theory Stirling imports Main begin 1.1 Stirling numbers of the second kind fun Stirling :: nat ⇒ nat ⇒ nat where Stirling 0 0 = 1 | Stirling 0 (S...