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With this paper we introduce a new series representation of $zeta(3)$, which is based on the Clausen representation of odd integer zeta values. Although, relatively fast converging series based on the Clausen representation exist for $zeta(3)$, 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 in combination with a polynomial description of the even Bernoulli numbers, the convergence behavior of the Clausen-type representation has been improved to a level that allows us to challenge ultimately all BBP-type formulas available for $zeta(3)$. We present an explicit numerical comparison between one of the best available BBP formulas and our formalism. Furthermore, we demonstrate by an explicit computation using the first four terms in our series representation only that $zeta(3)$ results with an accuracy of $2*10^{-26}$, where our computation guarantees on each approximation level for an analytical expression for $zeta(3)$.
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_{
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of
We introduce the degenerate Bernoulli numbers of the second kind as a degenerate version of the Bernoulli numbers of the second kind. We derive a family of nonlinear differential equations satisfied by a function closely related to the generating fun
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We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.