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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 | Abstract: | We 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 | Services: | Forum | Review | PDF | Favorites |
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