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

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, , and the Legendre chi function, , defined by

and

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.

**1.**K. M. Dempsey, D. Liu and J. P. Dempsey,*Plana's summation formula for , , ,*, Math. Comp.**55**(1990), 693-703. MR**91b:65003****2.**W. Gautschi,*On certain slowly convergent series occurring in plate contact problems*, Math. Comp.**57**(1991), 325-338. MR**91j:40002****3.**J. Boersma and J. P. Dempsey,*On the numerical evaluation of Legendre's chi-function*, Math. Comp.**59**(1992), 157-163. MR**92k:65008****4.**D. Cvijovic and J. Klinowski,*Closed-form summation of some trigonometric series*, Math. Comp.**64**(1995), 205-210. MR**95f:65017****5.**H. J. Weaver,*Theory of discrete and continuous Fourier analysis*, John Wiley, New York, 1989. MR**90c:42002****6.**W. Magnus, F. Obergettinger and R. P. Soni,*Formulas and theorems for the special functions of mathematical physics*, Springer-Verlag, Berlin, 1966. MR**38:1291****7.**L. Lewin,*Polylogarithms and associated functions*, North-Holland, Amsterdam, 1981. MR**83b:33019****8.**M. Abramowitz and I. Stegun (eds.),*Handbook of mathematical functions with formulas, graphs and mathematical tables*, U. S. Government Printing Office, 1966. MR**34:8607****9.**D. Cvijovic and J. Klinowski,*New formulae for the Bernoulli and Euler polynomials at rational arguments*, Proc. Amer. Math. Soc.**123**(1995), 1527-1535. MR**95g:11085**

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

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