Probabilistic Heterogeneous Stirling Numbers

• The paper introduces a unified framework that generalizes classical, degenerate, and probabilistic families through explicit generating functions and recurrence relations.
• It establishes Dobiński-type identities and summation formulas that interconnect Bell, Stirling, and Lah polynomials, enabling smooth interpolation between deterministic and stochastic models.
• It leverages probabilistic settings, including Poisson and Bernoulli cases, to bridge combinatorial enumeration with moment-based statistical interpretations.

Probabilistic heterogeneous Bell polynomials unify and extend several classical and probabilistic families of combinatorial polynomials, including Bell, Stirling, Lah, and Lah–Bell polynomials. By incorporating both a probabilistic structure—via a random variable $Y$ satisfying specific moment conditions—and a continuous degeneracy or "heterogeneity" parameter $\lambda\in\mathbb R\backslash\{0\}$, these polynomials form a flexible family which smoothly interpolates between classical, degenerate, and random-weight combinatorial structures. The underlying framework synthesizes the heterogeneous approach of Kim and Kim with generalized probabilistic tools to yield explicit generating functions, summation formulas, Dobiński-type identities, rich recurrence structures, and connections to partial Bell polynomials and well-known distributions (Kim et al., 15 Jan 2026).

1. Definition and Generating Function

Let $Y$ be a real random variable with $E[\,|Y|^n\,]<\infty$ for all $n\ge0$ and $\lim_{n\to\infty}|t|^n E[|Y|^n]/n! = 0$ for $|t|<r$. Define i.i.d. copies $\{Y_j\}_{j\ge1}$ and $S_k = Y_1 + \cdots + Y_k$. The degenerate exponential $e_\lambda^z(t)$ is given by

$$e_\lambda^z(t) = \sum_{m=0}^\infty \frac{\langle z\rangle_{m,\lambda}}{m!}\,t^m,\quad \langle z\rangle_{m,\lambda} = z(z+\lambda)\cdots(z+(m-1)\lambda).$$

The probabilistic heterogeneous Bell polynomials $\{H_{n,\lambda}^Y(x)\}_{n\ge0}$ are defined by the exponential generating function

$$\exp\bigl\{x\bigl(E[e_\lambda^{-Y}(-t)]-1\bigr)\bigr\} = \sum_{n=0}^\infty H_{n,\lambda}^Y(x) \frac{t^n}{n!}.$$

This generalizes several well-known generating functions: classical Bell polynomials ($\lambda \to 0,\ Y \equiv 1$), heterogeneous Bell polynomials ($Y \equiv c$), and probabilistic Bell polynomials ($\lambda \to 0,\ Y$ arbitrary) (Kim et al., 15 Jan 2026, Kim et al., 30 Mar 2025, Kim et al., 2024).

2. Explicit Summation Formulas and Structure Constants

$H_{n,\lambda}^Y(x)$ admits the expansion

$$H_{n,\lambda}^Y(x) = \sum_{k=0}^n x^k\, H_\lambda^Y(n,k)$$

where the probabilistic heterogeneous Stirling numbers of the second kind are given by

$$H_\lambda^Y(n,k) = \frac{1}{k!}\sum_{j=0}^k \binom{k}{j}(-1)^{k-j} E\bigl[\langle S_j\rangle_{n,\lambda}\bigr].$$

An alternative involves a double sum: $$H_\lambda^Y(n,k) = \sum_{l=k}^n {\,l\brack k\,}_Y\, [n\brack l]\; \lambda^{n-l}$$ with ${l\brack k\,}_Y$ denoting probabilistic Stirling numbers (second kind) and $[n\brack l]$ unsigned classical Stirling numbers of the first kind (Kim et al., 15 Jan 2026). This algebraic structure enables direct interpolation between Stirling and Lah families and their probabilistic analogues.

3. Dobiński-type Identity and Probabilistic Interpretation

The family $H_{n,\lambda}^Y(x)$ possesses a Dobiński-type infinite sum: $$H_{n,\lambda}^Y(x) = e^{-x}\;\sum_{k=0}^\infty \frac{E\bigl[\langle S_k\rangle_{n,\lambda}\bigr]}{k!}\,x^k.$$ This directly recovers the classical Dobiński formula in the limits $\lambda\to0$ and $Y$ deterministic, and the Lah–Bell case when $\lambda\to1$. By analogy with the degenerate moment representation for heterogeneous Bell polynomials, this identity shows that for $Y$ Poisson, $E[\langle S_k\rangle_{n,\lambda}]$ is a deterministic function of $k$. Thus, $H_{n,\lambda}^Y(x)$ encodes the summation of (generalized) falling factorial moments associated to random partitions and weighted block structures (Kim et al., 15 Jan 2026, Kim et al., 30 Mar 2025).

4. Recurrence Relations and Convolution Identities

The polynomials $H_{n,\lambda}^Y(x)$ satisfy the following recurrences:

• First-order recurrence:

$$H_{n+1,\lambda}^Y(x) = x\sum_{k=0}^n \binom{n}{k}\, E[\langle Y\rangle_{k+1,\lambda}]\, H_{n-k,\lambda}^Y(x).$$

• Binomial convolution:

$$H_{n,\lambda}^Y(x+y) = \sum_{k=0}^n \binom{n}{k} H_{k,\lambda}^Y(x)\, H_{n-k,\lambda}^Y(y).$$

• Derivative relations:

$$\frac{d^k}{dx^k}H_{n,\lambda}^Y(x) = k!\sum_{j=0}^{n-k}\binom{n}{j} H_{j,\lambda}^Y(x)H_{\lambda}^Y(n-j,k).$$

In particular,

$$\frac{d}{dx}H_{n,\lambda}^Y(x) = \sum_{j=0}^{n-1}\binom{n}{j} E[\langle Y\rangle_{n-j,\lambda}] H_{j,\lambda}^Y(x).$$

These relations generalize classical identities for Bell and Stirling numbers to the full heterogeneous probabilistic setting. The underlying structure supports equational manipulation, explicit enumeration, and analytic continuation in $\lambda$ (Kim et al., 15 Jan 2026).

5. Connections to Partial Bell Polynomials

The link to partial Bell polynomials $B_{n,k}(x_1,\ldots,x_{n-k+1})$, which enumerate the number of partitions of a set with prescribed block sizes, is realized by

$$H_{n,\lambda}^Y(x) = \sum_{k=0}^n x^k\;B_{n,k}\bigl( E[\langle Y\rangle_{1,\lambda}],\, E[\langle Y\rangle_{2,\lambda}],\,\ldots, E[\langle Y\rangle_{n-k+1,\lambda}] \bigr).$$

Moreover,

$$H_{\lambda}^Y(n,k) = B_{n,k}\bigl( E[\langle Y\rangle_{1,\lambda}],\ldots, E[\langle Y\rangle_{n-k+1,\lambda}]\bigr).$$

This embedding allows for systematically transferring analytic and combinatorial results across the heterogeneous, probabilistic, and classical domains using partial Bell polynomial identities as a bridge (Kim et al., 15 Jan 2026).

6. Specializations to Poisson and Bernoulli Distributions

Explicit computations for canonical choices of $Y$ yield:

• Poisson($\alpha$) case:

$$E[\langle S_k\rangle_{n,\lambda}] = H_{n,\lambda}(k\alpha), \quad H_{n,\lambda}^Y(x) = \sum_{k=0}^n \phi_k(x)\, \alpha^k\, H_{\lambda}(n,k),$$

where $\phi_k(x)$ are classical Bell polynomials.

• Bernoulli($p$) case:

$$E[\langle Y\rangle_{n,\lambda}] = p\,\langle 1\rangle_{n,\lambda},\;\; H_\lambda^Y(n,k) = p^k H_\lambda(n,k),\;\; H_{n,\lambda}^Y(x) = H_{n,\lambda}(x\,p).$$

In these cases, the probabilistic structure collapses or reduces to a deterministic deformation, simplifying analysis yet providing links to classical probability and combinatorics (Kim et al., 15 Jan 2026).

7. Unification and Interpolation: Classical, Degenerate, and Probabilistic Families

The probabilistic heterogeneous Bell polynomials interpolate continuously between

• Classical Bell polynomials ($\lambda\to0,\ Y\equiv 1$)
• Heterogeneous Bell polynomials ($Y\equiv c$)
• Lah–Bell polynomials ($\lambda=1,\ Y\equiv 1$)
• Probabilistic Bell and Lah–Bell polynomials ($\lambda\to 0$, $Y$ arbitrary; $\lambda=1$, $Y$ arbitrary)

As $\lambda$ varies from $0$ to $1$ and the distribution of $Y$ is chosen accordingly, the full combinatorial content of classical, degenerate, and weighted families is subsumed. This parameter-driven perspective facilitates analytical approaches to interpolation, moment-cumulant relations, and asymptotic enumeration, as well as offering a unified framework for recurrence relations, generating functions, Dobinski-type sums, and explicit enumeration formulas (Kim et al., 15 Jan 2026, Kim et al., 30 Mar 2025, Kim et al., 2024).

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A plausible implication is that this heterogeneous probabilistic paradigm may serve as the foundation for further extensions to other combinatorial sequences and analytical structures, connecting algebraic, probabilistic, and enumerative combinatorics.