Remote Access Mathematics of Computation
Green Open Access

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
Full-text PDF

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}


\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.

References [Enhancements On Off] (What's this?)

  • 1. K. M. Dempsey, D. Liu and J. P. Dempsey, Plana's summation formula for $\sum m^{-2}\sin(m\alpha)$, $m^{-3}\cos(m\alpha)$, $m^{-2}A^m$, $m^{-3}A^m$, 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

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65B10, 11M35

Retrieve articles in all journals with MSC (1991): 65B10, 11M35

Additional Information

Djurdje Cvijovic
Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

Jacek Klinowski
Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

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

American Mathematical Society