This article gives a direct formula for the computation of B(n) using the asymptotic formula $$B (n) \approx 2 {\frac {n!}{π^{n}{2}^{n}}}$$ where n is even and $n >> 1$. This is simply based on the fact that $ζ(n)$ is very near 1 when n is large and since $B (n) = 2 {\frac {ζ(n) n!}{π^{n}{2}^{n}}}$ exactly. The formula chosen for the Zeta function is the one with prime numbers from the well-known Euler product for $ζ(n)$. This algorithm is far better than the recurrence formula for the Bernoulli numbers even if each B(n) is computed individually. The author could compute $B (750,000)$ in a few hours. The current record of computation is now (as of Feb. 2007) $B (5,000,000)$ a number of (the numerator) of 27332507 decimal digits is also based on that idea.
We present a computer algebra approach to proving identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the required identities without computing the integrals.
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 computation of $B_{2n}$ as a function of B$_{2n-2}$ only. Furthermore, we show that a direct computation of the Riemann zeta-function and their derivatives at k $\in \mathbb Z$ is possible in terms of these polynomial representation. As an explicit example, our polynomial Bernoulli number representation is applied to fast approximate computations of $ζ$(3), $ζ$(5) an…
We describe an algorithm for computing Bernoulli numbers. Using a parallel implementation, we have computed B(k) for k = 10^8, a new record. Our method is to compute B(k) modulo p for many small primes p, and then reconstruct B(k) via the Chinese Remainder Theorem. The asymptotic time complexity is O(k^2 log(k)^(2+epsilon)), matching that of existing algorithms that exploit the relationship between B(k) and the Riemann zeta function. Our implementation is significantly faster than several existing implementations of the zeta-function method.
We consider the computation of Bernoulli, Tangent (zag), and Secant (zig or Euler) numbers. In particular, we give asymptotically fast algorithms for computing the first n such numbers in O(n^2.(log n)^(2+o(1))) bit-operations. We also give very short in-place algorithms for computing the first n Tangent or Secant numbers in O(n^2) integer operations. These algorithms are extremely simple, and fast for moderate values of n. They are faster and use less space than the algorithms of Atkinson (for Tangent and Secant numbers) and Akiyama and Tanigawa (for Bernoulli numbers).
In this work we study the properties of a new algebraic variant of the degenerate Bernoulli polynomial $\tildeβ_{k}(m,x)$ and study the corresponding degenerate Bernoulli number $\tildeβ_{k}(m,1)=m^{k}β_{k}(1/m)$, where $β_{k}(λ), λ\neq 0$ is the standard degenerate Bernoulli number. Our approach relies on a new algebraic framework for generating functions and the action of a symbolic evaluation function on powers of polynomials. We show that $\tildeβ_{k}(m,x)$ displays surprising links with other mathematical objects (such as Circulant matrices and Galois fields) and enjoys many interesting a…
In this note, starting with a little-known result of Kuo, I derive a recurrence relation for the Bernoulli numbers $B_{2 n}$, $n$ being any positive integer. This new recurrence seems advantageous in comparison to other known formulae since it allows the computation of both $B_{4 n}$ and $B_{4 n +2}$ from only $B_0, B_2,..., B_{2n}$.
With this paper we introduce a new series representation of $ζ(3)$, which is based on the Clausen representation of odd integer zeta values. Although, relatively fast converging series based on the Clausen representation exist for $ζ(3)$, their convergence behavior is very slow compared to BBP-type formulas, and as a consequence they are not used for explicit numerical computations. The reason is found in the fact that the corresponding Clausen function can be calculated analytically for a few rational arguments only, where $x=\frac{1}{6}$ is the smallest one. Using polylogarithmic identities …
We study matrices which transform the sequence of Fibonacci or Lucas polynomials with even index to those with odd index and vice versa. They turn out to be intimately related to generalized Stirling numbers and to Bernoulli, Genocchi and tangent numbers and give rise to various identities between these numbers. There is also a close connection with the Akiyama-Tanigawa algorithm.
In this paper we investigate special generalized Bernoulli polynomials with a,b,c parameters that generalize classical Bernoulli numbers and polynomials. The present paper deals with some recurrence formulae for the generalization of poly-Bernoulli numbers and polynomials with a,b,c parameters. Poly-Bernoulli numbers satisfy certain recurrence relationships which are used in many computations involving poly-Bernoulli numbers. Obtaining a closed formula for generalization of poly-Bernoulli numbers with a,b,c paramerers therefore seems to be a natural and important problem. By using the generali…
We develop a method for generating pseudorandom binary sequences using the Bernoulli map on cubic algebraic integers. The distinguishing characteristic of our generator is that it generates chaotic true orbits of the Bernoulli map by exact computation. In particular, we clarify a way to properly prepare a set of initial points (i.e., seeds), which is needed when generating multiple pseudorandom sequences. With this seed selection method, we can distribute the initial points almost uniformly in the unit interval and can also guarantee that the orbits starting from them do not merge. We also rep…
Let $β\in(1,2)$ be a Pisot number and let $H_β$ denote Garsia's entropy for the Bernoulli convolution associated with $β$. Garsia, in 1963 showed that $H_β<1$ for any Pisot $β$. For the Pisot numbers which satisfy $x^m=x^{m-1}+x^{m-2}+...+x+1$ (with $m\ge2$) Garsia's entropy has been evaluated with high precision by Alexander and Zagier and later improved by Grabner, Kirschenhofer and Tichy, and it proves to be close to 1. No other numerical values for $H_β$ are known. In the present paper we show that $H_β>0.81$ for all Pisot $β$, and improve this lower bound for certain ranges of $β$. Our me…
The aim of this paper is to construct generating functions for some families of special finite sums with the aid of the Newton-Mercator series, hypergeometric series, and $p$-adic integral (the Volkenborn integral). By using these generating functions, their functional equations, and their partial derivative equations, many novel computational formulas involving the special finite sums of (inverse) binomial coefficients, the Bernoulli type polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the (alternating) harmonic numbers, the Leibnitz polynomials and others. Among…
Let $B_n$ ($n = 0, 1, 2, ...$) denote the usual $n$-th Bernoulli number. Let $l$ be a positive even integer where $l=12$ or $l \geq 16$. It is well known that the numerator of the reduced quotient $|B_l/l|$ is a product of powers of irregular primes. Let $(p,l)$ be an irregular pair with $B_l/l \not\equiv B_{l+p-1}/(l+p-1) \modp{p^2}$. We show that for every $r \geq 1$ the congruence $B_{m_r}/m_r \equiv 0 \modp{p^r}$ has a unique solution $m_r$ where $m_r \equiv l \modp{p-1}$ and $l \leq m_r < (p-1)p^{r-1}$. The sequence $(m_r)_{r \geq 1}$ defines a $p$-adic integer $χ_{(p, l)}$ which is a zer…
We prove certain general forms of functional relations among Witten multiple zeta-functions in several variables (or zeta-functions of root systems). The structural background of those functional relations is given by the symmetry with respect to Weyl groups. From those relations we can deduce explicit expressions of values of Witten zeta-functions at positive even integers, which is written in terms of generalized Bernoulli numbers of root systems. Furthermore we introduce generating functions of those Bernoulli numbers of root systems, by which we can give an algorithm of calculating Bernoul…
For the computation of terms of the Baker-Campbell-Hausdorff series $H = \log(e^Ae^B})$ some a priori knowledge about the denominators of the coefficients of the series can be beneficial. In this paper an explicit formula for the computation of common denominators for the rational coefficients of the homogeneous components of the series is derived. Explicit computations up to degree 30 show that the common denominators obtained by this formula are as small as possible, which suggests that the formula is in a sense optimal. The sequence of integers defined by the formula seems to be interesting…
