2-adic partial Stirling functions and their zeros

Let $P_n(x) = \frac{1}{n!} \sum \binom{n}{2i+1}^x$. 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 $e \geq 2$ and $2^e + 1 \leq n \leq 2^e + 4$, then $P_n$ has exactly $2^{e-1}$ zeros, one in each $\mathrm{mod} \; 2^{e-1}$ congruence. We study the relationship between the zeros of $P_{2^e + \Delta}$ and $P_\Delta$, for $1 \leq \Delta \leq 2^e$, and the convergence of $ P_{2^e + \Delta} (x)$ as $e \to \infty$.

Keywords

Stirling number, 2-adic integers

2010 Mathematics Subject Classification

Primary 11B73. Secondary 05Axx.

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Published 29 October 2014