On the expansion of the power of any polynomial 1+x+x^2+x^3+x^4+etc

Thw purpose of this paper is to present a systemic study of some families of the generalized q-Euler numbers and polynomials of higher order.

In this note, we first review the novel approach to power sums put forward recently by Muschielok in arXiv:2207.01935v1, which can be summarized by the formula $S_m^{(a)}(n) = \sum_{k} c_{mk} \psi_k^{(a)}(n)$, where the $c_{mk}$'s are the expansion coefficients and where the basis functions $\psi_m^{(a)}(n)$ fulfil the recursive property $\psi_m^{(a+1)}(n)= \sum_{i=1}^n \psi_m^{(a)}(i)$. Then, we point out a number of supplementary facts concerning the said approach not conte...

In this paper, some formulae for Genoochi polynomials of higher order are derived using the fact that sets of Bernoulli and Euler polynomials of higher order form basis for the polynomial space.

We pose the question of what is the best generalization of the factorial and the binomial coefficient. We give several examples, derive their combinatorial properties, and demonstrate their interrelationships. On cherche ici \`a d\'eterminer est la meilleure g\'en\'eralisation possible des factorielles et des coefficients du bin\^oome. On s'interesse \`a plusieurs exemples, \`a leurs propri\'et\'es combinatoires, et aux differentes relations qu'ils mettent en jeu.

This is the classical monograph on the combinatorial study of Eulerian polynomials, published in 1970. It has been retyped in TeX and made available on the web with the kind permission of Springer-Verlag. This on-line version has an ouput of 49 pages. Written in French it contains the following items: 0. Introduction to and review of the Euler Numbers 1. General properties of the systems of exceedances and rises 2. The Eulerian polynomials 3. The exponential formula 4. Genera...

In this paper, we discuss sums of powers of the positive integers and compute both the exponential and ordinary generating functions for these sums. We express these generating functions in terms of exponential and geometric polynomials and also show their connection to other interesting series. In particular, we show their connection to an interesting problem of Ovidiu Furdui.

In this paper we use computational method based on operational point of view to prove a new generating function of exponential polynomials. We give its applications involving geometric polynomials, Bernoulli and Euler numbers.

Binomial Theorem for (N+n)^r is described with non-commuting variables N and n.

Euler gives a continued fraction representation of (1+x)^n involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,... and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fraction at z=t sqrt(-1), for ``vanishing'' n, and for infinite n.

Translation from the Latin original, "Demonstratio gemina theorematis Neutoniani, quo traditur relatio inter coefficientes cuiusvis aequationis algebraicae et summas potestatum radicum eiusdem" (1747). E153 in the Enestrom index. In this paper Euler gives two proofs of Newton's identities, which express the sums of powers of the roots of a polynomial in terms of its coefficients. The first proof takes the derivative of a logarithm. The second proof uses induction and the fact...