Related papers: Identities between polynomials related to Stirling…
We consider sequences of polynomials that satisfy differential-difference recurrences. Polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete…
We introduce the $B$-Stirling numbers of the first and second kind, which are the coefficients of the potential polynomials when we express them in terms of the monomials and the falling factorials, respectively. These numbers include, as…
In this paper we consider the Daehee numbers and polynomials of the first and second kind, and give several explicit representations for them. In particular, we express the Daehee polynomials as the derivative of a generalized binomial…
Two doubly indexed families of homogeneous and isobaric polynomials in several indeterminates are considered: the (partial) exponential Bell polynomials $B_{n,k}$ and a new family $S_{n,k}$ such that $X_1^{-(2n-1)}S_{n,k}$ and $B_{n,k}$…
We introduce a class of $f(t)$-factorials, or $f(t)$-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the…
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
We offer several new summation identities involving harmonic numbers, odd harmonic numbers, and Fibonacci numbers. Our results are derived using three different approaches: partial summation, polynomial identities and binomial…
In this paper, we derive a formula for the sums of powers of the first $n$ positive integers, $S_k(n)$, that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the…
In this paper, we provide some novel binomial convolution related to symmetric functions, as well as convolution sums without the binomial symbol. Moreover we give some new convolution sums of Bernoulli, Euler, and Genocchi numbers and…
In this paper we study q-Bernoulli numbers and polynomials related to q-Stirling numbers. From thsese studying we investigate some interesting q-stirling numbers' identities related to q-Bernoulli numbers.
Several new estimates for the 2-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as…
In this paper, we consider the poly-Bernoulli numbers and polynomials of the second kind and presents new and explicit formulae for calculating the poly-Bernoulli numbers of the second kind and the Stirling numbers of the second kind.
We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa-Kaneko zeta function, complementing and generalizing earlier…
In this paper, we continue our investigation of double sums where the inner sum is binomial but incomplete. We prove many new results for these types of double sums associated with binomial transform pairs. As applications we deduce new…
In this work we prove a new combinatorial identity and applying it we establish many finite harmonic sum identities. Among many others, we prove that \begin{equation*}…
For the Stirling numbers of the second kind $S(n,k)$ and the ordered Bell numbers $B(n)$, we prove the identity $\sum_{k=1}^{n/2} S(n,2k)(2k-1)! = B(n-1)$. An analogous identity holds for the sum over odd $k$'s.
Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integers $S_k^{(m)}(n)$ \begin{equation*} S_k^{(m)}(n) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i…
During the course of an ongoing work on the small-$x$ behaviour of parton distribution functions, some identities have been found which involve Stirling numbers of the first and the second kind, as well as binomial coefficients. Without any…
The aim of this paper is to study a dimorphic property associated with two different sums of identically independent Bernoulli random variables having two different families of probability mass functions. In addition, we give two…
We present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. \begin{align*} \sum_{k=0}^{n}\binom{\alpha}{n-k}\binom{\beta+k}{k}x^k…