Related papers: Power sum identities with generalized Stirling num…
In this paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.
We consider two types of polynomials $F_n (x) = \sum_{\nu=1}^n \nu! S_2(n,\nu) x^\nu$ and $\hat{F}_n (x) = \sum_{\nu=1}^n \nu! S_2(n,\nu) H_\nu x^\nu$, where $S_2(n,\nu)$ are the Stirling numbers of the second kind and $H_\nu$ are the…
In this paper, we obtain formulas for the number of representations of positive integers as sums of arbitrarily many squares (and other polygonal numbers) with a certain natural weighting. The resulting weighted sums give Fourier…
Let Y be a random variable satisfying specific moment conditions. This paper introduces and investigates probabilistic heterogeneous Stirling numbers of the second kind and probabilistic heterogeneous Bell polynomials. These structures…
We prove astonishing identities generated by compositions of positive integers. In passing, we obtain two new identities for Stirling numbers of the first kind. In the two last sections we clarify an algebraic sense of these identities and…
We elaborate on the recent suggestion to consider averaging of Cauchy identities for the Schur functions over power sum variables. This procedure has apparent parallels with the Borel transform, only it changes the number of combinatorial…
A systematic procedure for generating certain identities involving elementary symmetric functions is proposed. These identities, as particular cases, lead to new identities for binomial and q-binomial coefficients.
By using the associated and restricted Stirling numbers of the second kind, we give some generalizations of the poly-Bernoulli numbers. We also study their arithmetical and combinatorial properties. As an application, at the end of the…
Recently, Dil and Boyadzhiev \cite{AD2015} proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence $\left( {{{\left\{ 0 \right\}}_r},1} \right)$. In this paper we show that the sums of multiple…
Given two combinatorial identities proved earlier, a new set of variations of these combinatorial identities is listed and proved with the integral representation method. Some identities from literature are shown to be special cases of…
In terms of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.
We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary…
Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…
We give combinatorial proofs of $q$-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitz's identity, a new proof of the $q$-Frobenius identity of Garsia and Remmel and of Ehrenborg's Hankel…
In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops.…
In this paper we give a convolution identity for the complete and elementary symmetric functions. This result can be used to proving and discovering some combinatorial identities involving $r$-Stirling numbers, $r$-Whitney numbers and…
We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.
In this paper, we derive novel formulas and identities connecting Cauchy numbers and polynomials with both ordinary and generalized Stirling numbers, binomial coefficients, central factorial numbers, Euler polynomials, $r$-Whitney numbers,…
We show how infinite series of a certain type involving generalized harmonic numbers can be computed using a knowledge of symmetric functions and multiple zeta values. In particular, we prove and generalize some identities recently…
We present several sequences involving harmonic numbers and the central binomial coefficients. The calculational technique is consists of a special summation method that allows, based on proper two-valued integer functions, to calculate…