2-进Stirling函数及其零点
2-adic Stirling functions and their zeros
摘要：
令$P_n(x)=\frac1{n!}\sum\binom n{2i+1}(2i+1)^x$。这扩展到一个连续的函数上的2-adic整数，PDMATHAMN第2-adic部分斯特林函数。证明了$(-1)^{n+1}P_n$是第二类Stirling数$S(x,n)$唯一的2-基连续逼近.我们提出了广泛的信息$P_n$的零点，其中有许多有趣的模式。证明了如果$e\ge2$和$2^e+1\le n\le 2^e+4$，则$P_n$恰好有$2^{e-1}$ 0，且每个模$2^{e-1}$同余中有1个0.本文研究了$P_{2^e+Δ}$和$P_Δ$的零点之间的关系，对于$1\leΔ\le 2^e$，以及$P_{2^e+Δ}(x)$收敛于$e\to\infty$ .
英文摘要：
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 $P_{2^e+Δ}$ and $P_Δ$, for $1\leΔ\le 2^e$, and the convergence of $P_{2^e+Δ}(x)$ as $e\to\infty$.
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