Advertisement

The Ramanujan Journal

, Volume 21, Issue 2, pp 123–143 | Cite as

Fractional sums and Euler-like identities

  • Markus Müller
  • Dierk Schleicher
Article

Abstract

We introduce a natural definition for sums of the form
$$\sum_{\nu=1}^xf(\nu)$$
when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the Γ function or Euler’s little-known formula \(\sum_{\nu=1}^{-1/2}\frac{1}{\nu}=-2\ln 2\) .
Many classical identities like the geometric series and the binomial theorem nicely extend to this more general setting. Sums with a fractional number of terms are closely related to special functions, in particular the Riemann and Hurwitz ζ functions. A number of results about fractional sums can be interpreted as classical infinite sums or products or as limits, including identities like
$$\begin{array}{l}\displaystyle\lim_{n\to\infty}\Biggl[e^{\frac{n}{4}(4n+1)}n^{-\frac{1}{8}-n(n+1)}(2\pi)^{-\frac{n}{2}}\prod_{k=1}^{2n}\Gamma\biggl(1+\frac{k}{2}\biggr)^{k(-1)^k}\Biggr]\\[12pt]\quad =\displaystyle\sqrt[12]{2}\exp\biggl(\frac{5}{24}-\frac{3}{2}\zeta'(-1)-\frac{7\zeta(3)}{16\pi^2}\biggr),\end{array}$$
some of which seem to be new; and even for those which are known, our approach provides a new method to derive these identities and many others.
Fractional sum Summation Interpolation Summation identities 

Mathematics Subject Classification (2000)

33B99 40C99 40A25 41A05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1974) Google Scholar
  2. 2.
    Adamchik, V.S.: The multiple Gamma function and its application to computation of series. Ramanujan J. 9(3), 271–288 (2005) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berndt, B.: Ramanujan’s Notebooks, Part I. Springer, New York (1989) Google Scholar
  4. 4.
    Borwein, P., Dykshoorn, W.: An interesting infinite product. J. Math. Anal. Appl. 179(1), 203–207 (1993) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Conway, J.B.: Functions of One Complex Variable I. Springer, New York (1978) Google Scholar
  6. 6.
    Euler, L.: Dilucidationes in capita postrema calculi mei differentialis de functionibus inexplicabilibus, 2nd edn. Commentatio 613 Indicis Enestroemiani. Mémoires de l’Académie des Sciences de St.-Petersbourg, vol. 4, pp. 88–119 (1813) Google Scholar
  7. 7.
    Gosper, R.W., Ismail, M.E.H., Zhang, R.: On some strange summation formulas. Ill. J. Math. 37(2), 240–277 (1993) MATHMathSciNetGoogle Scholar
  8. 8.
    Müller, M., Schleicher, D.: How to add a non-integer number of terms, and how to produce unusual infinite summations. J. Comput. Appl. Math. 178(1–2), 347–360 (2005) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Müller, M., Schleicher, D.: How to add a non-integer number of terms: from axioms to new identities. Manuscript (in preparation) Google Scholar
  10. 10.
    Spanier, J., Oldham, K.B.: An Atlas of Functions. Springer, Berlin (1987) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Research I/MathematicsJacobs UniversityBremenGermany

Personalised recommendations