Values of the Legendre chi and Hurwitz zeta functions at rational arguments

Cvijović, Djurdje; Klinowski, Jacek

doi:10.1090/S0025-5718-99-01091-1

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Values of the Legendre chi
and Hurwitz zeta functions
at rational arguments

Authors: Djurdje Cvijovic and Jacek Klinowski
Journal: Math. Comp. 68 (1999), 1623-1630
MSC (1991): Primary 65B10; Secondary 11M35
Posted: May 17, 1999
MathSciNet review: 1648375
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the Hurwitz zeta function, $\zeta(\nu,a)$ , and the Legendre chi function, $\chi _\nu(z)$ , defined by

$\begin{displaymath}\zeta(\nu,a)=\sum _{k=0}^\infty\frac{1}{(k+a)^\nu},\quad 0<a\le 1,\operatorname{Re}\,\nu>1,\end{displaymath}$

and

$\begin{displaymath}\chi _\nu(z)=\sum _{k=0}^\infty \frac{z^{2k+1}}{(2k+1)^\nu},\quad|z|\le 1,\operatorname{Re}\,\nu>1\ \text{with}\ \nu=2,3,4,\dotsc,\end{displaymath}$

respectively, form a discrete Fourier transform pair. Many formulae involving the values of these functions at rational arguments, most of them unknown, are obtained as a corollary to this result. Among them is the further simplification of the summation formulae from our earlier work on closed form summation of some trigonometric series for rational arguments. Also, these transform relations make it likely that other results can be easily recovered and unified in a more general context.

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