Let $p_k(n)$ be the number of partitions of $n$ with exactly $k$ parts. It is known that
$$ f_k(x)=\sum_n p_k(n)x^n=x^k\prod_{i=1}^k\frac{1}{1-x^i}.$$
My question is: is there a known analytic continuation of the function $f_k(x)$?
Analytic continuation of partitions generating function
complex-analysisnumber theory
Best Answer
The product formula is an analytic continuation of $f_k$. Note that in the product formula, $f_k$ is a finite product of meromorphic functions on $\mathbb{C}$, so it is meromorphic on $\mathbb{C}$ (and therefore analytic on every point that is not a pole). However, it’s important to note that the full partitions generating function $$f(x)=\sum_{n=0}^{\infty}p(x)x^n=\prod_{n=1}^{\infty}\frac{1}{1-x^n}$$ does not have an analytic continuation because it has a singularity at every root of unity, which are dense in the unit circle. Thus, it is lacunary.