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

Authors:
Djurdje Cvijović and Jacek Klinowski

Journal:
Math. Comp. **68** (1999), 1623-1630

MSC (1991):
Primary 65B10; Secondary 11M35

DOI:
https://doi.org/10.1090/S0025-5718-99-01091-1

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 \[ \zeta (\nu ,a)=\sum _{k=0}^\infty \frac {1}{(k+a)^\nu },\quad 0<a\le 1,\operatorname {Re} \nu >1,\] and \[ \chi _\nu (z)=\sum _{k=0}^\infty \frac {z^{2k+1}}{(2k+1)^\nu },\quad |z|\le 1,\operatorname {Re} \nu >1 \text {with} \nu =2,3,4,\dotsc ,\] 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 Cvijović**

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

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