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

Euler investigates the Taylorseries of (1+x+xx)^n and uses the results to evaluates some integrals which are today often proved with the calculus of residues.

In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The so-called binomial sums are also considered. The problem of constructing polynomials that allow to calculate the values of the corresponding sums in certain cases is solved.

In this paper, we study polynomials of the form $f(x)=(x^n+x^{n-1}+...+1)^l$ for $l=1,2,3,4$ to generate a pattern titled "unique coefficient pattern". Namely, we analyze each unique coefficient patterns of $f(x)$ and generate functions titled "relation functions". The approach that we follow will allow us to evaluate desired coefficients for such polynomial expansions by simply using these relation functions.

What is a general expression for the sum of the first n integers, each raised to the mth power, where m is a positive integer? Answering this question will be the aim of the paper....We will take the unorthodox approach of presenting the material from the point of view of someone who is trying to solve the problem himself. Keywords: analogy, Johann Faulhaber, finite sums, heuristics, inductive reasoning, number theory, George Polya, problem solving, teaching of mathematics

In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials. Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore seems to be a natural and important problem.

We show that explicit forms for certain polynomials~$\psi^{(a)}_m(n)$ with the property \[ \psi^{(a+1)}_m(n) = \sum_{\nu=1}^n \psi_m^{(a)}(\nu) \] can be found (here, $a,m,n\in\mathbb{N}_0$). We use these polynomials as a basis to express the monomials~$n^m$. Once the expansion coefficients are determined, we can express the $m$-th power sums~$S^{(a)}_m(n)$ of any order $a$, \[ S^{(a)}_m(n) = \sum_{\nu_a = 1}^n \cdots \sum_{\nu_2 = 1}^{\nu_3} \sum_{\nu_1=1}^{\nu_2} \nu_1^m, \...

The purpose of this paper is to present a syatemic study of some familes of higher-order Euler numbers and polynomials. In particular, by using the basis property of higher-order Euler polynomials for the space of polynomials of degree less than and equal to n, we derive some interesting identities for the higher-order Euler polynomias.

Translated from the Latin original, "Theorema arithmeticum eiusque demonstratio", Commentationes arithmeticae collectae 2 (1849), 588-592. E794 in the Enestroem index. For m distinct numbers a,b,c,d,...,\upsilon,x this paper evaluates \[ \frac{a^n}{(a-b)(a-c)(a-d)...(a-x)}+\frac{b^n}{(b-a)(b-c)(b-d)...(b-x)} +...+\frac{x^n}{(x-a)(x-b)(x-c)...(x-\upsilon)}. \] When $n \leq m-2$, the sum is 0, which Euler had already shown in sect. 1169 of his Institutiones calculi integralis, ...

By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several identities and summation formul\ae\ parallel to those of the usual binomial coefficients.

This paper, along with E592 and E636, seems to consider the binomial expansion (1+z)^n in the case where z is complex. Euler even gives the sums of divergent series. The paper is translated from Euler's Latin original into German.