Related papers: Series for even powers of Pi by generalization Eul…
Translated from the Latin original, "Observationes circa bina biquadrata quorum summam in duo alia biquadrata resolvere liceat" (1772). E428 in the Enestroem index. This paper is about finding A,B,C,D such that $A^4+B^4=C^4+D^4$. In sect.…
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.
The problem of finding formulas for sums of powers of natural numbers has been of interest to mathematicians for many centuries. Among these is Faulhaber's well-known formula expressing the power sums as polynomials whose coefficients…
The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years the subject has found important applications in the modelling of…
We introduce a series of numbers which serve as a generalization of Bernoulli, Euler numbers and binomial coefficients. Their properties are applied to solve a probability problem and suggest a statistical test for independence and…
Euler--Maclaurin and Poisson analogues of the summations $\sum_{a < n \leq b} \chi(n) f(n)$, $\sum_{a < n \leq b} d(n) f(n)$, $\sum_{a < n \leq b} d(n) \chi (n) f(n)$ have been obtained in a unified manner, where $(\chi (n))$ is a periodic…
Although many series exist for $\pi$ and $\pi^2$, very few are known for $\pi^3$. In this article, we derive, using a trigonometric identity obtained by Euler, two representations of $\pi^3$ involving infinite sums and the golden ratio. The…
This translation has been published in Stephen Hawking (ed.), "God Created the Integers", published in 2007 by Running Press. There may have been some changes to the final published version and this copy. This is a translation from the…
This paper introduces a symbolic calculus-based approach for deriving closed-form expressions for the sums of arithmetic sequences. The method extends beyond constant-difference sequences to those with polynomially increasing steps,…
In modern usage the Bernoulli numbers and Bernoulli polynomials follow Euler's approach and are defined using generating functions. We consider the functional equation $f(x)+x^k=f(x+1)$ and show that a solution can be derived from…
In this paper we propose a method for proving some exponential inequalities based on power series expansion and analysis of derivations of the corresponding functions. Our approach provides a simple proof and generates a new class of…
In $1735$ Euler \cite{1} proved that for each positive integer $k$, the series $\zeta(2k) = \sum_{\ell=1}^{\infty} \ell^{-2k}$ converges to a rational multiple of $\pi^{2k}$. Many demonstrations of this fact are now known, and Euler's…
Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the $r$-fold summation of $1^m,2^m,...,n^m$ is a polynomial in $n(n+r)$ when $m$ is a positive odd number. The present…
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…
We extend several celebrated methods in classical analysis for summing series of complex numbers to series of complex matrices. These include the summation methods of Abel, Borel, Ces\'aro, Euler, Lambert, N\"orlund, and Mittag-Leffler,…
We provide a multidimensional weighted Euler--MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences…
In recent, H. Sun defined a new kind of refined Eulerian polynomials, namely, \begin{eqnarray*} A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)} \end{eqnarray*} for $n\geq 1$, where ${odes}(\pi)$ and ${edes}(\pi)$…
Euler explored the problem of finding three numbers such that the sum or difference of any two of them is a perfect square. He discovered a parametric solution represented by polynomials of degree 18 and identified the smallest of these…
We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate…
We show how to find series expansions for $\pi$ of the form $\pi=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}$, where S(n) is some polynomial in $n$ (depending on $m,p,a$). We prove that there exist such expansions for $m=8k$, $p=4k$,…