Fractional Sums and Euler-like Identities

Abstract

We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] 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 gamma 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 zeta functions. A number of results about fractional sums can be interpreted as classical infinite sums or products or as limits, including identities like \[ \lim_{n\to\infty}[ e^{\frac n 4(4n+1)}n^{-\frac 1 8 - n(n+1)}(2\pi)^{-\frac n 2} \prod_{k=1}^{2n} \Gamma(1+\frac k 2)^{k(-1)^k} ] = \sqrt[12]{2} \exp({5/24} - \frac 3 2 \zeta'(-1) -\frac{7\zeta(3)}{16\pi^2}) \] some of which seem to be new.