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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
Published electronically: May 17, 1999
MathSciNet review: 1648375
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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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Additional Information

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: http://dx.doi.org/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
Published electronically: May 17, 1999
Article copyright: © Copyright 1999 American Mathematical Society