Well I've always liked the idea of having a sequence \( a_n \) and finding an analytic continuation \( a(s) \) such that it returns \( a_n \) at integer values. Recently I have found a method of continuum summing http://math.eretrandre.org/tetrationforu...hp?tid=801

This is one of the exploits we have.

We will be using two theorems, my continuum sum theorem and a result thanks to Grunwald and Letnikov.

\( \frac{d^{-s}f(t)}{dt^{-s}} = \lim_{n\to \infty} n^{s}\sum_{k=0}^{\infty} \frac{s \cdot \cdot \cdot (s + k -1)}{k!} f(t - \frac{k}{n}) \)

Which reduces to the natural derivative and integral for integer values of s

We take our sequence \( a_n \) and create an auxillary function \( \vartheta(s) = \sum_{n=0}^{\infty} a_n \frac{s^n}{n!} \)

We are clear that

\( \frac{d^n}{dt^n}\vartheta(t)|_{t=0} = a_n \)

And therefore if we create the following function:

\( \phi(s) = \int_0^\infty e^{-t} \frac{d^{-s}\vartheta(t)}{dt^{-s}}dt \)

We have, thanks to my continuum sum theorem:

\( \phi(-n) - \phi(-n-1) = a_n \)

Therefore the function:

\( \phi(-s) - \phi(-s-1) = a(s) \) and interpolates the sequence \( a_n \)

And even more beautifully:

\( \phi(-n) - \phi(-n-k) = \sum_{j=n}^{n+k-1} a_j \)

and so we have also a continuum sum from phi.

This is a nifty little way of creating interpolation functions. Thought it was neat. Allows us to interpolate something like the prime number sequence and get an entire function that returns prime numbers at integer values. this is because the primenumber sequence doesn't grow too fast and so our auxillary function is entire.

This is one of the exploits we have.

We will be using two theorems, my continuum sum theorem and a result thanks to Grunwald and Letnikov.

\( \frac{d^{-s}f(t)}{dt^{-s}} = \lim_{n\to \infty} n^{s}\sum_{k=0}^{\infty} \frac{s \cdot \cdot \cdot (s + k -1)}{k!} f(t - \frac{k}{n}) \)

Which reduces to the natural derivative and integral for integer values of s

We take our sequence \( a_n \) and create an auxillary function \( \vartheta(s) = \sum_{n=0}^{\infty} a_n \frac{s^n}{n!} \)

We are clear that

\( \frac{d^n}{dt^n}\vartheta(t)|_{t=0} = a_n \)

And therefore if we create the following function:

\( \phi(s) = \int_0^\infty e^{-t} \frac{d^{-s}\vartheta(t)}{dt^{-s}}dt \)

We have, thanks to my continuum sum theorem:

\( \phi(-n) - \phi(-n-1) = a_n \)

Therefore the function:

\( \phi(-s) - \phi(-s-1) = a(s) \) and interpolates the sequence \( a_n \)

And even more beautifully:

\( \phi(-n) - \phi(-n-k) = \sum_{j=n}^{n+k-1} a_j \)

and so we have also a continuum sum from phi.

This is a nifty little way of creating interpolation functions. Thought it was neat. Allows us to interpolate something like the prime number sequence and get an entire function that returns prime numbers at integer values. this is because the primenumber sequence doesn't grow too fast and so our auxillary function is entire.