关于Lehmer提出的Comtet数问题
On a Question of Lehmer concerning the Comtet Numbers
AI总结：
本文将Comtet数视为加权斯特林多项式的特殊值，证明其满足四项递推式，回答了Lehmer关于$B(n,k)$是否存在固定项数递推的问题，并推导了Touchard型多项式的相关关系。
AI中文摘要：
第一类和第二类Comtet数$b(n,k)$与$B(n,k)$分别源于$(1+x)\log(1+x)$及其复合逆的幂次。将两类数视为加权斯特林多项式的特殊值，我们证明四项递推式$kB(n+1,k+1)=B(n,k-1)-(n-k)B(n,k)-B(n+1,k)$，回答Lehmer关于$B(n,k)$是否满足固定项数递推的问题。OEIS A354794中M. Kurkov曾推测无符号数组的对应关系但未证明，我们还推导了两类数对应的Touchard型多项式的显式公式、卷积恒等式及微分-差分关系。
英文摘要：
The Comtet numbers $b(n,k)$ and $B(n,k)$ of the first and second kind arise from the powers of $(1+x)\log(1+x)$ and of its compositional inverse, respectively. By interpreting both families as special values of weighted Stirling polynomials, we answer Lehmer's question of whether $B(n,k)$ satisfies a recurrence with a fixed number of terms by proving the four-term recurrence $kB(n+1,k+1)=B(n,k-1)-(n-k)B(n,k)-B(n+1,k)$. The corresponding relation for the unsigned array was conjectured by M. Kurkov, but not proved, in OEIS A354794. We further derive explicit formulas, convolution identities, and differential-difference relations for the Touchard-type polynomials attached to the two families.
