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$m$-Bell and $m$-Stirling numbers: Iterated binomial transforms, hyper-Bessel functions, and moments of the Conway--Maxwell--Poisson distribution — arXiv:2608.12011

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.12011
License (as archived): CC0-1.0
Archived: 2026-10-19T05:47:29.813Z
SHA-256 of the archived text: d1f7fbddb9cddfb61c337267969fe8ec6a6245587c71386eee80faac912cedb2
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How to cite

$m$-Bell and $m$-Stirling numbers: Iterated binomial transforms, hyper-Bessel functions, and moments of the Conway--Maxwell--Poisson distribution — arXiv:2608.12011 — arXiv (en), https://arxiv.org/abs/2608.12011 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_e032a2da89a4 (Metraipolis archive dc_e032a2da89a4, sha256 d1f7fbddb9cddfb61c337267969fe8ec6a6245587c71386eee80faac912cedb2)

BibTeX

@misc{dce032a2da89a4,
  title = {$m$-Bell and $m$-Stirling numbers: Iterated binomial transforms, hyper-Bessel functions, and moments of the Conway--Maxwell--Poisson distribution — arXiv:2608.12011},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.12011},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_e032a2da89a4; sha256 d1f7fbddb9cddfb61c337267969fe8ec6a6245587c71386eee80faac912cedb2},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.12011. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\nu=m$: 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-13 in math.PR).
Categorie arXiv: Combinatorics (math.CO) ; Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2608.12011 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-19T05:40:02.435Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-13?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).