$m$-Bell and $m$-Stirling numbers: Iterated binomial transforms, hyper-Bessel functions, and moments of the Conway--Maxwell--Poisson distribution

Abstract

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.

Disclosure

“, modified Bessel functions, hypergeometric functions. Computational results and other supporting information are available on GitHub. Claude Fable 5 (Anthropic) assisted with some of the derivations; its contributions are documented in the repository’s pull requests, whereas the human author’s contributions are direct commits, so the repository history”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:11:27.496876Z

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