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Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
A novel power series representation of the generalized Marcum $Q-$function of positive order involving generalized Laguerre polynomials is presented. The absolute convergence of the proposed power series expansion is showed, together with a…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
In this paper, we prove Newton-Maclaurin type inequalities for functions obtained by linear combination of two neighboring primary symmetry functions, which is a generalization of the classical Newton-Maclaurin inequality.
The generalized definition of symmetry is formulated. Application of this definition for symmetric analysis of theoretical physics equations is considered. The version of electrodynamics is constructed permitting the faster-than-light…
In this paper, we consider mixed sums of generalized polygonal numbers. Specifically, we obtain a finiteness condition for universality of such sums; this means that it suffices to check representability of a finite subset of the positive…
In this paper, we introduce The 2-variable unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials and derive some implicit summation formulae and general symmetry identities. The result extend some known summations…
The paper explores connection between the generalized order and the generalized type of an entire function and the speed of the best polynomial approximation in the unit disk. The relations which define the generalized order and the…
In this paper we focus on two new families of polynomials which are connected with exponential polynomials and geometric polynomials. We discuss their generalizations and show that these new families of polynomials and their generalizations…
The objective of this paper is to derive some interesting properties of Genocchi, Euler and Bernstein polynomials by means of the orthogonality of Hermite polynomials.
In this article we prove an algebraic identity which significantly generalizes the formula for sum of powers of consecutive integers involving Stirling numbers of the second kind. Also we have obtained a generalization of Newton-Girard…
In this note, we introduce generalized powers of linear operators. More precisely, operators are not raised to numbers but to other operators. We discuss several properties as regards this notion.
In this paper, we employ methods of contour integration and residue calculus to investigate the parity of two classes of cyclotomic Euler-type sums. One class involves products of cyclotomic harmonic numbers, while the other involves…
Taking inspiration from the work of Lanphier \cite{LANPHIER2022125716}, a generalized multivariable polynomial formulation for sums of alternating powers is given, as well as analogous sums. Furthermore, an analog of the Euler-Maclaurin…
The main goal of this paper is to generalize Bohr's phenomenon from complex one-dimensional analysis to higher dimensions in the framework of Quaternionic Analysis.
We compute the (primary) equivariant Euler characteristics of the building for the general linear group over a finite field.
In this paper, we introduce a new type of generalized alternating hyperharmonic numbers $H_n^{(p,r,s_{1},s_{2})}$, and show that Euler sums of the generalized alternating hyperharmonic numbers $H_n^{(p,r,s_{1},s_{2})}$ can be expressed in…
In the present article, we study Bell based Euler polynomial of order {\alpha} and investigate some useful correlation formula, summation formula and derivative formula. Also, we introduce some relation of string number of the second kind.…
In this paper we present a generalization of Faulhaber's formula to sums of arbitrary complex powers $m\in\mathbb{C}$. These summation formulas for sums of the form $\sum_{k=1}^{\lfloor x\rfloor}k^{m}$ and $\sum_{k=1}^{n}k^{m}$, where…
In this article, we introduce congruential Euler numbers, which are a further generalization of generalized Euler numbers. We prove the $p$-adic congruences of congruential Euler numbers, which include answers to a conjecture related to…