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The computation of $\zeta(2k)$, $\beta(2k+1)$ and beyond by using telescoping series
- Publication Year :
- 2023
-
Abstract
- We present some simple proofs of the well-known expressions for \[ \zeta(2k) = \sum_{m=1}^\infty \frac{1}{m^{2k}}, \qquad \beta(2k+1) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^{2k+1}}, \] where $k = 1,2,3,\dots$, in terms of the Bernoulli and Euler polynomials. The computation is done using only the defining properties of these polynomials and employing telescoping series. The same method also yields integral formulas for $\zeta(2k+1)$ and $\beta(2k)$. In addition, the method also applies to series of type \[ \sum_{m\in\mathbb{Z}} \frac{1}{(2m-\mu)^s}, \qquad \sum_{m\in\mathbb{Z}} \frac{(-1)^m}{(2m+1-\mu)^s}, \] in this case using Apostol-Bernoulli and Apostol-Euler polynomials.<br />Comment: 24 pages, to appear in "Orthogonal Polynomials and Special Functions: In Memory of Jos\'e Carlos Petronilho''
- Subjects :
- Mathematics - Number Theory
Primary 40C15, Secondary 11M06
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2307.08063
- Document Type :
- Working Paper