A simpler proof of a Katsurada's theorem and rapidly converging series for $$\zeta {(2n+1)}$$ ζ ( 2 n + 1 ) and $$\beta {(2n)}$$ β ( 2 n )
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| 024 | 7 | 0 | |a 10.1007/s10231-014-0409-3 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s10231-014-0409-3 | ||
| 100 | 1 | |a Lima |D F. |u Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasília, DF, Brazil |4 aut | |
| 245 | 1 | 2 | |a A simpler proof of a Katsurada's theorem and rapidly converging series for $$\zeta {(2n+1)}$$ ζ ( 2 n + 1 ) and $$\beta {(2n)}$$ β ( 2 n ) |h [Elektronische Daten] |c [F. Lima] |
| 520 | 3 | |a In a recent work on Euler-type formulae for even Dirichlet beta values, i.e., $$\beta {(2n)}$$ β ( 2 n ) , I have derived an exact closed-form expression for a class of zeta series. From this result, I have conjectured closed-form summations for two families of zeta series. Here in this work, I begin by using a known formula by Wilton to prove those conjectures. As example of applications, some special cases are explored, yielding rapidly converging series representations for the Apéry constant, $$\zeta (3)$$ ζ ( 3 ) , and the Catalan constant, $$G = \beta (2)$$ G = β ( 2 ) . Interestingly, our series for $$\zeta (3)$$ ζ ( 3 ) converges faster than that used by Apéry in his irrationality proof (Astérisque 61:11-13, 1979). Also, our series for $$G$$ G converges faster than a celebrated one discovered by Ramanujan (J Indian Math Soc VII:93-96, 1915). At last, I present a simpler, more direct proof for a recent theorem by Katsurada which generalizes the above results. | |
| 540 | |a Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 | ||
| 690 | 7 | |a Riemann zeta function |2 nationallicence | |
| 690 | 7 | |a Dirichlet beta function |2 nationallicence | |
| 690 | 7 | |a Zeta series |2 nationallicence | |
| 690 | 7 | |a Clausen function |2 nationallicence | |
| 773 | 0 | |t Annali di Matematica Pura ed Applicata (1923 -) |d Springer Berlin Heidelberg |g 194/4(2015-08-01), 1015-1024 |x 0373-3114 |q 194:4<1015 |1 2015 |2 194 |o 10231 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s10231-014-0409-3 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s10231-014-0409-3 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 100 |E 1- |a Lima |D F. |u Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasília, DF, Brazil |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Annali di Matematica Pura ed Applicata (1923 -) |d Springer Berlin Heidelberg |g 194/4(2015-08-01), 1015-1024 |x 0373-3114 |q 194:4<1015 |1 2015 |2 194 |o 10231 | ||