Fractional Sums and Euler-like Identities

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 ∑ −1/2 ν=1 1 ν = −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