Mathematics of Computation

Author(s): Djurdje Cvijovic; Jacek Klinowski.
Journal: Math. Comp. 68 (1999), 1623-1630.
MSC (1991): Primary 65B10; Secondary 11M35
Posted: May 17, 1999
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Abstract: We show that the Hurwitz zeta function, $\zeta(\nu,a)$ , and the Legendre chi function, $\chi _\nu(z)$ , defined by

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

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

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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Djurdje Cvijovic
Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
Email: dc133@cam.ac.uk

Jacek Klinowski
Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
Email: jk18@cam.ac.uk

DOI: 10.1090/S0025-5718-99-01091-1
PII: S 0025-5718(99)01091-1
Keywords: Summation of series, Hurwitz's zeta function, Legendre's chi function
Received by editor(s): February 16, 1998
Posted: May 17, 1999
Copyright of article: Copyright 1999, American Mathematical Society

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