Related papers: On an infinite number of nonlinear Euler sums
In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum…
In the last decades many authors have become interested in the study of multilinear and polynomial generalizations of families of operator ideals (such as, for instance, the ideal of absolutely summing operators). However, these…
A novel approach is introduced for deriving exact solutions to nonlinear systems of ordinary differential equations. This method consists of four parts. In the initial part, the examined nonlinear differential equation system is transformed…
In this work, series expansions in terms of Bessel functions of the first kind are given for the sine and cosine integrals. These representations differ from many of the known Neumann-type series expansions for the sine and cosine…
We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power…
The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum $\sum_{k=0}^{n-1} f(k)$ of values of a function $f$ by a linear combination of a corresponding integral of…
By extending a construction due to Gross and McMullen [2], we show that for any odd integer n and for any even integer d>n+2 there are infinitely many Salem numbers $\alpha$ of degree d such that $\alpha^n-1$ is a unit. A similar result is…
A new one-parameter family of iterative method for solving nonlinear equations is constructed and studied. Two variants, both with cubic convergence, are developed, one for finding simple zeros and other for multiple zeros of known…
In this paper, we present several explicit formulas of the sums and hyper-sums of the powers of the first (n+1)-terms of a general arithmetic sequence in terms of Stirling numbers and generalized Bernoulli polynomials.
We present algorithms to evaluate two types of multiple sums, which appear in higher-order loop computations. We consider expansions of a generalized hypergeometric-type sums, $\sum_{n_1,...,n_N} [Gamma(a1.n+c1) Gamma(a2.n}+c2) ...…
Translated from the Latin original "Evolutio producti infiniti $(1-x)(1-xx)(1-x^3)(1x^4)(1-x^5)(1-x^6)$ etc. in seriem simplicem" (1775). E541 in the Enestroem index. In this paper Euler is revisiting his proof of the pentagonal number…
Our work is motivated by the fact that the norms of the Eulerian integers are related to the sums of form $a^2-ab+b^2$, providing a natural generalization for problems concerning products over sums or differences of integers. Let $E$ be the…
Spectral method related to Lame equation with finite-gap potential is used to study the optical cascading equations. These equations are known not to be integrable by inverse scattering method. Due to "partial integrability" two-gap…
Let $$\lambda(s)=\sum_{n=0}^\infty\frac1{(2n+1)^s},$$ $$\beta(s)=\sum_{n=0}^\infty\frac{(-1)^{n}}{(2n+1)^s},$$ and $$\eta(s)=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}$$ be the Dirichlet lambda function, its alternating form, and the Dirichlet…
Recently de Thanhoffer de V\"olcsey and Van den Bergh classified the Euler forms on a free abelian group of rank 4 having the properties of the Euler form of a smooth projective surface. There are two types of solutions: one corresponding…
We will employ the method of contour integration to investigate the parity results of non-embedded cyclotomic multiple $t$-values, which we refer to as cyclotomic Euler $T$-sums. We can provide explicit parity formulas for the linear and…
In this paper we present two different results in the context of nonlinear analysis. The first one is essentially a nonlinear technique that, in view of its strong generality, may be useful in different practical problems. The second…
We define the generalized-Euler-constant function $\gamma(z)=\sum_{n=1}^{\infty} z^{n-1} (\frac{1}{n}-\log \frac{n+1}{n})$ when $|z|\leq 1$. Its values include both Euler's constant $\gamma=\gamma(1)$ and the "alternating Euler constant"…
In 1946, Erd\"os and Niven proved that no two partial sums of the harmonic series are equal. In this paper, we extend this result by demonstrating that no two partial sums of the hyperharmonic series are equal.
In terms of the derivative operator, integral operator and Saalsch\"{u}tz's theorem, two families of summation formulae involving generalized harmonic numbers are established.