My favorite integer sequence

[jaydfox]

Deriving the continuous function is actually pretty straightforward.

(...)

In general:
\(
D\left[b_k x^k\right] = b_{k-1} \left(\frac{x}{2}\right)^{k-1} \\
\Rightarrow k b_k x^{k-1} = \frac{1}{2^{k-1}} b_{k-1} x^{k-1} \\
\Rightarrow b_k = \frac{1}{k} \left(\frac{1}{2^{k-1}}\right) b_{k-1}
\)

By induction:
\(
b_k = \frac{1}{2^{(k-1)k/2} k!}
\)

Then, as I posted previously:

\(
f(x) = \sum_{k=0}^{\infty}\frac{1}{2^{k(k-1)/2} k!} x^k
\)

I'm kind of disappointed I didn't see this earlier. I didn't even arrive at it by "first principles". Rather, I was playing around with creating a function that obeys the functional equation f'(x) = f(x/2), but which oscillates in a similar manner to the way the binary partition function oscillates. It's actually pretty cool, especially the code I came up with. Perhaps I'll share in a later post.

At any rate, it dawned on me that the oscillating version of the function behaves like a Laurent series, with positive powers of x creating oscillations far from the origin (with a periodicity of about two zeroes per doubling of scale), and negative powers of x creating oscillations near the origin (with a periodicity of about one zero per doubling of scale).

And as I was trying to formalize it, I realized something that I'd missed earlier, which should have been obvious:

\(
f(x) = \sum_{k=0}^{\infty}\frac{1}{2^{k(k-1)/2} k!} x^k \\
\Rightarrow f(x) = \sum_{k=0}^{\infty}\frac{1}{2^{k(k-1)/2} \Gamma(k+1)} x^k
\)

Treating Gamma(k) at negative integers as infinity, and the reciprocal of such as zero, we can take the limit from negative to positive infinity. And we can replace k with (k+b), where b is zero in the original solution, but can now be treated as any real (well, any complex number, but the complex versions are less interesting).

\(
f_{\beta}(x) = \sum_{k=-\infty}^{\infty}\frac{2^{-(k+\beta)(k+\beta-1)/2}}{\Gamma(k+\beta+1)} x^{k+\beta}
\)

So, without further ado, here are some graphs comparing f_{1/2} to f_0: