That was my secret reason for asking. A formula I found in 1964 for summing at least negative powers of the "integers" from 1 to x (not necessarily an integer) is: F(x; s) = (1/Gamma(s)) Integral{0 <= t <= 1} (1 - t^x) / (1 - t) (-ln(t))^(s-1) dt = Sum_{1 <= n <= x} 1/n^s . (Letting x -> oo gives an integral formula for the product of Gamma(s) and Zeta(s). When I mentioned this to my freshman advisor, Henry McKean, he showed me was already in an old book, A Course of Modern Analysis by Whittaker and Watson -- to my great dismay.) I never studied how far F(x; s) can be analytically continued. Maybe it comes around and works for negative s -- giving a formulas for partial sums of positive powers of integers??? --Dan
On Jan 8, 2015, at 10:14 AM, Steve Witham <sw@tiac.net> wrote:
Can't these be used to sum real, complex and negative powers (except -1) of sequences of integers, as with Faulhaber's formula? For "sequence lengths" that are real, negative or complex, too? All the variations of Faulhaber's formula seem to break down when the power is -1, but in just that case there's a generalization of the harmonic numbers H(n) for complex n. --Steve
From: Daniel Asimov <asimov@msri.org> Thanks, Mike. That's a very nice paper! --Dan
On Jan 6, 2015, at 12:04 PM, Mike Stay<metaweta@gmail.com> wrote:
http://arxiv.org/abs/physics/9705021
On Tue, Jan 6, 2015 at 11:16 AM, Daniel Asimov<dasimov@earthlink.net> wrote:
Are there extensions of Bernoulli numbers to real (or complex) index?
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