Related papers: Binomial sums related to rational approximations t…

In this paper, we present new explicit simultaneous rational approximations converging sub-exponentially to the values of Bell polynomials at the points of the form $(\gamma, 1! (2a+1)\zeta(2), 2!\zeta(3),..., (m-1)!(a+1+(-1)^ma)\zeta(m)),$…

Already in 1734 Euler found a short explicit formula for the value of Riemann zeta function Zeta(s) when the argument s equals a positive integer 2n where n=1,2,3,. No such formula exists for odd positive integer arguments of Zeta. The…

In this work we introduce a new polynomial representation of the Bernoulli numbers in terms of polynomial sums allowing on the one hand a more detailed understanding of their mathematical structure and on the other hand provides a…

In this paper, we prove two related central binomial series identities: $B(4)=\sum_{n \geq 0} \frac{\binom{2n}n}{2^{4n}(2n+1)^3}=\frac{7 \pi^3}{216}$ and $C(4)=\sum_{n \in \mathbb{N}} \frac{1}{n^4 \binom{2n}n}=\frac{17 \pi^4}{3240}.$ Both…

In 1978, Apery has given sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery's proof are second-order difference equations with polynomial…

The author derives new family of series representations for the values of the Riemann Zeta function $\zeta(s)$ at positive odd integers. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with…

We obtain two sequences of rational numbers which converge to the Euler-Gompertz constant. Denote by <f(x)> the integral of f(x)e^{-x} from 0 to infinity. Recall that the Euler-Gompertz constant \delta is <ln(x+1)>. Main idea. Let P_n(x) be…

Defining a Beukers [1] like integral for $\zeta(5)$ as \begin{equation*} I_n:=\int_{(0,1)^5}\frac{(1-x_3)^n(1-x_4)^n P_n(x_1)P_n(x_2)}{1-(1-x_1x_2x_3x_4)x_5} \ dx_1dx_2dx_3dx_4dx_5 \end{equation*} we prove that for each $n\in\mathbb{N}$…

In this shortnote, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values $\:\zeta{(2 k +1)}$, $\zeta{(s)}$ being the Riemann zeta function and $k$ a positive integer, is…

New (infinitely many) rational approximants to \zeta(3) proving its irrationality are given. The recurrence relations for the numerator and denominator of these approximants as well as their continued fraction expansions are obtained. A…

This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…

Let $S := \{p_1,\ldots ,p_{\ell}\}$ be a finite set of primes and denote by $\mathcal{U}_S$ the set of all rational integers whose prime factors are all in $S$. Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order…

We obtain an asymptotic series $\sum_{j=0}^\infty\frac{I_j}{n^j}$ for the integral $\int_0^1[x^n+(1-x)^n]^{\frac1{n}}dx$ as $n\to\infty$, and compute $I_j$ in terms of alternating (or "colored") multiple zeta value. We also show that $I_j$…

Tuenter [Fibonacci Quarterly 40 (2002), 175-180] and other authors have considered centred binomial sums of the form \[S_r(n) = \sum_k \binom{2n}{k}|n-k|^r,\] where $r$ and $n$ are non-negative integers. We consider sums of the form…

Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for zeta(4n+3) which generalizes Apery's series for zeta(3), and appears to give the best possible series…

In 2007, A.I.Aptekarev and his collaborators discovered a sequence of rational approximations to Euler's constant $\gamma$ defined by a linear recurrence. In this paper, we generalize this result and present an explicit construction of…

We observe that five polynomial families have all of their zeros on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers,…

The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on the real…

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive, square-free discriminant and $p$ be a fixed prime number. In this paper, we have shown the finiteness result for the solutions of the Diophantine…

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lison\v{e}k states that inserting all cyclic shifts of some fixed blocks of 2's into the multiple zeta value {\zeta}(1,3,...,1,3) gives an explicit rational multiple of a…