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29 March 2024
 
  » arxiv » math.CA/0502109

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Fractional Sums and Euler-like Identities
Markus Müller ; Dierk Schleicher ;
Date 5 Feb 2005
Subject Classical Analysis and ODEs MSC-class: 33-99, 40-99, 40A25, 41A05 | math.CA
AbstractWe introduce a natural definition for sums of the form sum_{ u=1}^x f( u) 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 $Gamma$ function or Euler’s little-known formula (sum_{ u=1}^{-1/2} frac 1 u = -2ln 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 $zeta$ functions. A number of results about fractional sums can be interpreted as classical infinite sums or products or as limits, including identities like prod_{ u=1}^infty frac 1 e (1+frac 1 {a u} )^{a u+frac 1 2} =sqrtfrac{Gamma(1+frac 1 a)}{2pi} exp[ {frac 1 2 (1+frac 1 a)-a(zeta’(-1,1+frac 1 a)-zeta’(-1))} ] some of which seem to be new.
Source arXiv, math.CA/0502109
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