K-function

Summary

The K-function extends the hyperfactorial $1^1 \cdot 2^2 \cdots n^n$ to non-integer arguments. It satisfies $K(1) = 1$ and $K(z + 1) = z^z K(z)$, so $K(n + 1) = 1^1 2^2 \cdots n^n$ for positive integers $n$. It is defined through an integral of $\log \Gamma$, where $\Gamma$ is the gamma function, and is related to the Barnes G-function.