11/19/2009, 10:28 PM

I just found a simple way to convert between a Taylor series at 0 and a Dirichlet series.

This requires a fractional differintegral.

For any function with :

Pretty cool, huh?

Using this conversion, the Riemann Zeta function can be expressed as:

which I believe is somewhat well known due to its connection with Bernoulli polynomials, and 1/(1 - e^z) is well known to be the generating function for Bernoulli polynomials. Still, I've never seen this formula before, it is interesting to see it like this.

This requires a fractional differintegral.

For any function with :

Pretty cool, huh?

Using this conversion, the Riemann Zeta function can be expressed as:

which I believe is somewhat well known due to its connection with Bernoulli polynomials, and 1/(1 - e^z) is well known to be the generating function for Bernoulli polynomials. Still, I've never seen this formula before, it is interesting to see it like this.