m-贝尔数与m-斯特林数：迭代二项式变换、超贝塞尔函数及康威-麦克斯韦-泊松分布的矩
$m$-Bell and $m$-Stirling numbers: Iterated binomial transforms, hyper-Bessel functions, and moments of the Conway--Maxwell--Poisson distribution
AI总结：
该研究推广贝尔数得到m-贝尔数，构造m-斯特林三角阵列并证明相关结构定理，还揭示m-贝尔数对应康威-麦克斯韦-泊松分布（ν=m）的矩，拓展了相关数论与概率分布的联系。
AI中文摘要：
我们引入了贝尔数的一种自然推广：m-贝尔数B^{(m)}_n，其特征为对其应用m次二项式变换后，可得到向左平移m位的原序列。它们的指数生成函数满足m阶常微分方程，其解为超几何（超贝塞尔）函数；当m=1时退化为指数函数（对应经典贝尔数），当m=2时退化为修正贝塞尔函数（得到“贝塞尔-贝尔”数）。仿照贝尔-斯特林对应关系，我们从两项递推式S_m(n+1,k)=m⌊k/m⌋S_m(n,k)+S_m(n,k-1)构造了m-斯特林三角阵列，并证明了一个基本平移恒等式，由此可推出中心结构定理：m-斯特林三角的行和重现B^{(m)}_n，更细致地，剩余类行和恰好是m个本原m-贝尔序列。m-斯特林数以对偶对形式存在（与第一类伴侣、广义下降阶乘及拉赫型伴侣配对），可作为多项式基之间的转换算子，具有类似多比斯基（Dobiński）的公式，还可计数瓮模型中受同余约束的分拆及受限排列插入历史。最后，我们证明m-贝尔数控制着离散参数ν=m为整数的康威-麦克斯韦-泊松分布的矩：归一化矩是固定超贝塞尔载体比的组合，其整数系数恰好是本原m-贝尔序列；当m=1时，这对应经典结论——泊松分布的矩即为贝尔数。
英文摘要：
We introduce a natural generalization of the Bell numbers: the $m$-Bell numbers $B^{(m)}_{n}$, characterized by the property that $m$ applications of the binomial transform reproduce the original sequence shifted $m$ places to the left. Their exponential generating functions satisfy $m$-th order ordinary differential equations whose solutions are hypergeometric (hyper-Bessel) functions, specializing to the exponential function when $m=1$ (classical Bell numbers) and to modified Bessel functions when $m=2$ (yielding "Bessel-Bell" numbers). Mirroring the Bell-Stirling correspondence, we construct $m$-Stirling triangular arrays from the two-term recurrence $S_m (n+1,k) = m \left\lfloor k/m \right\rfloor S_m(n,k)+S_m(n,k-1)$ and prove an elementary shift identity from which the central structure theorem follows: the row sums of the $m$-Stirling triangle reproduce $B^{(m)}_{n}$, and, more finely, the residue-class row sums are precisely the $m$ primitive $m$-Bell sequences. The $m$-Stirling numbers come in dual pairs (with first-kind partners, generalized falling factorials, and Lah-type companions), serve as conversion operators between polynomial bases, admit Dobiński-like formulas, and count congruence-constrained partitions in an urn model as well as restricted permutation insertion histories. Finally, we show that the $m$-Bell numbers govern the moments of the Conway-Maxwell-Poisson distribution with integer dispersion parameter $ν=m$: the scaled moments are combinations of fixed hyper-Bessel carrier ratios whose integer coefficients are precisely the primitive $m$-Bell sequences, recovering for $m=1$ the classical fact that the moments of the Poisson distribution are the Bell numbers.
