AI 中文总结
针对Subbarao和Verma1999年提出的互补贝尔数相关公开问题，本文结合有限差分法、部分Motzkin路径等方法，证明了互补贝尔数仅对任意给定值取有限次的结论。
AI 中文摘要
1999年，Subbarao和Verma提出了关于互补贝尔数序列$(f(n))_{n \geq 0}$的若干公开问题，该序列可通过贝尔多项式$B_{n}(x) = \sum_{k=0}^{n} \left\{ \begin{smallmatrix} n \\\\ k \end{smallmatrix} \right\} x^k$定义，其中$f(n) = B_{n}(-1)$。Yang随后于2001年在《Electron. J. Combin.》上解决了Subbarao和Verma提出的前两个问题，但据我们所知，第三个问题仍未解决。第三个问题的第一部分询问$f(n)$是否仅对任意给定值取有限次，我们通过综合应用基于有限差分的方法、部分Motzkin路径、Tate代数关于高斯范数的完备性以及Strassmann定理，肯定地解决了该问题。
英文摘要
Subbarao and Verma introduced, in 1999, a number of open problems concerning the sequence $(f(n))_{n \geq 0}$ of complementary Bell numbers, which may be defined via Bell polynomials $B_{n}(x) = \sum_{k=0}^{n} \left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\} x^k$ so that $f(n) = B_{n}(-1)$. Yang [Electron. J. Combin., 2001] subsequently solved the first two problems from Subbarao and Verma, but the third such problem has remained open, to the best of our knowledge. The first part of this third problem asks whether or not $f(n)$ takes any given value only a finite number of times. We solve this problem in the affirmative, through a combined application of finite difference-based methods, partial Motzkin paths, the completeness of the Tate algebra with respect to the Gauss norm, and Strassmann's theorem.
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