2-adic Stirling functions and their zeros

Abstract

Let $P_n(x)=\frac1{n!}\sum\binom n{2i+1}(2i+1)^x$. This extends to a continuous function on the 2-adic integers, the $n$th 2-adic partial Stirling function. We show that $(-1)^{n+1}P_n$ is the only 2-adically continuous approximation to $S(x,n)$, the Stirling number of the second kind. We present extensive information about the zeros of $P_n$, for which there are many interesting patterns. We prove that if $e\ge2$ and $2^e+1\le n\le 2^e+4$, then $P_n$ has exactly $2^{e-1}$ zeros, one in each mod $2^{e-1}$ congruence. We study the relationship between the zeros of $P_{2^e+\Delta}$ and $P_\Delta$, for $1\le\Delta\le 2^e$, and the convergence of $P_{2^e+\Delta}(x)$ as $e\to\infty$.