Fractional sums and Euler-like identities

We introduce a natural definition for sums of the form Sigma(x)(v=1)f(v) 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 Sigma(-1/2)(v=1) 1/v =-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 [GRAPHICS] 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.