Fractional sums and Euler-like identities

Abstract

We introduce a natural definition for sums of the form

$$\sum_{\nu=1}^xf(\nu)$$

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 Γ 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 ζ functions. A number of results about fractional sums can be interpreted as classical infinite sums or products or as limits, including identities like

$$\begin{array}{l}\displaystyle\lim_{n\to\infty}\Biggl[e^{\frac{n}{4}(4n+1)}n^{-\frac{1}{8}-n(n+1)}(2\pi)^{-\frac{n}{2}}\prod_{k=1}^{2n}\Gamma\biggl(1+\frac{k}{2}\biggr)^{k(-1)^k}\Biggr]\\[12pt]\quad =\displaystyle\sqrt[12]{2}\exp\biggl(\frac{5}{24}-\frac{3}{2}\zeta'(-1)-\frac{7\zeta(3)}{16\pi^2}\biggr),\end{array}$$

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.