Generalized Cosecant Numbers and the Hurwitz Zeta Function
| Autor: | Kowalenko, Victor |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | This announcement paper summarises recent development concerning the generalized cosecant numbers $c_{\rho,k}$, which represent the coefficients of the power series expansion for the important fundamental function $z^{\rho}/\sin^{\rho} z$. These coefficients are obtained for all, including complex, values of $\rho$ via the partition method for a power series expansion, which is more versatile than the standard Taylor series approach, but yields the same results as the latter when both can be applied, though in a different form. Surprisingly, the generalized cosecant numbers are polynomials in $\rho$ of degree $k$, where $k$ is the power of $z$. General formulas for the coefficients of the highest order terms in the generalized cosecant numbers are presented. It is then shown how the generalized cosecant numbers are related to the specific symmetric polynomials from summing over quadratic powers of integers. Consequently, integral values of the Hurwitz zeta function for even powers are expressed for the first time ever in terms of ratios of the generalized cosecant numbers. Comment: This 9-page paper was rejected by Electronic Research Announcements of the AMS on the grounds every reference is to a paper by the author. The conclusion was that the author is doing unmotivated work in complete isolation from the rest of the mathematical community, which is not something that should be encouraged |
| Databáze: | arXiv |
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