Mansour Toufik, Rastegar Reza, Roitershtein Alexander, Shattuck Mark
Department of Mathematics, University of Haifa, 3498838 Haifa, Israel.
Occidental Petroleum Corporation, 77046 Houston, TX.
Quaest Math. 2022;45(1):55-69. doi: 10.2989/16073606.2020.1848936. Epub 2020 Dec 5.
In this paper, we consider extensions of Spivey's Bell number formula wherein the argument of the polynomial factor is translated by an arbitrary amount. This idea is applied more generally to the -Whitney numbers of the second kind, denoted by (), where some new identities are found by means of algebraic and combinatorial arguments. The former makes use of infinite series manipulations and Dobinski-like formulas satisfied by (), whereas the latter considers distributions of certain statistics on the underlying enumerated class of set partitions. Furthermore, these two approaches provide new ways in which to deduce the Spivey formula for (). Finally, we establish an analogous result involving the -Lah numbers wherein the order matters in which the elements are written within the blocks of the aforementioned set partitions.
在本文中，我们考虑斯皮维贝尔数公式的扩展，其中多项式因子的自变量被任意量平移。这个想法更广泛地应用于第二类(q)-惠特尼数，记为(W^{(q)}(n,k))，通过代数和组合论证找到了一些新的恒等式。前者利用无穷级数运算以及(W^{(q)}(n,k))满足的类似多宾斯基公式，而后者考虑在基础的集合划分枚举类上某些统计量的分布。此外，这两种方法提供了推导(W^{(q)}(n,k))的斯皮维公式的新途径。最后，我们建立了一个涉及(q)-拉赫数的类似结果，其中元素在上述集合划分的块内书写的顺序很重要。
