Probabilistic Heterogeneous Bell Polynomials

• Probabilistic heterogeneous Bell polynomials are defined using random variable moments and a heterogeneity parameter to generalize classical combinatorial families such as Stirling, Lah, and Bell polynomials.
• They yield explicit summation formulas, Dobiński-like infinite series, and recurrence relations that continuously interpolate between classical and degenerate polynomial sequences.
• Their framework applies to Poisson and Bernoulli distributions, offering powerful analytical tools for enumeration and moment analysis in combinatorial research.

Probabilistic heterogeneous Bell polynomials generalize numerous classical combinatorial polynomial families by merging a probabilistic moment structure with a parameterized heterogeneity. Central to this theory is a random variable $Y$ satisfying specific moment conditions, along with a real heterogeneity parameter $\lambda\neq0$. The construction unifies Stirling, Lah, Bell, and their probabilistic analogues, producing explicit summation formulas, Dobiński-like infinite series, and recurrence identities, and recovers classical objects as limiting cases. The formalism integrates the moment framework of Kim and Kim with heterogeneous (degenerate) exponentials and provides applications to Poisson and Bernoulli distributions (Kim et al., 15 Jan 2026).

1. Foundational Definition and Generating Function

Given a random variable $Y$ with $E[|Y|^n]<\infty$ for all $n \ge 0$ and the analytic condition $\lim_{n\to\infty} \frac{|t|^n E[|Y|^n]}{n!}=0$ for $|t|< r$, let $\{Y_j\}_{j\ge1}$ denote i.i.d. copies and $S_k = Y_1 + \cdots + Y_k$. The probabilistic heterogeneous Bell polynomials $\{H_{n,\lambda}^Y(x)\}_{n\ge0}$ are defined by the exponential generating function: $\lambda\neq0$0 where the degenerate exponential $\lambda\neq0$1 is

$\lambda\neq0$2

This definition interpolates between classical and Lah-type Bell polynomials depending on the value of $\lambda\neq0$3.

2. Explicit Summation and Alternative Forms

The $\lambda\neq0$4th probabilistic heterogeneous Bell polynomial decomposes as

$\lambda\neq0$5

where the associated probabilistic heterogeneous Stirling numbers are given by

$\lambda\neq0$6

An equivalent double-sum representation is

$\lambda\neq0$7

with $\lambda\neq0$8 the probabilistic Stirling numbers of the second kind and $\lambda\neq0$9 unsigned Stirling numbers of the first kind. This structure unifies multiple classical combinatorial arrays within the same parameterized framework.

3. Dobiński-Type Identity and Limiting Behavior

A Dobinski-type infinite sum expresses $Y$0 via falling-factorial moments: $Y$1 This formula recovers the classical probabilistic Bell polynomials $Y$2 in the limit $Y$3 and the Lah–Bell polynomials $Y$4 for $Y$5, demonstrating that the $Y$6 parameter continuously interpolates between the main combinatorial sequences (Kim et al., 15 Jan 2026).

4. Recurrence Relations and Binomial Identities

The polynomials obey diverse recurrence relations:

• Time recurrence (order $Y$7):

$Y$8

• Binomial convolution:

$Y$9

• Derivative identities:

$E[|Y|^n]<\infty$0

and in particular

$E[|Y|^n]<\infty$1

These relations generalize known Bell and Stirling recurrences and encode the hierarchical combinatorial structure within the probabilistic heterogeneous framework.

5. Connection to Partial Bell Polynomials

The links to partial Bell polynomials $E[|Y|^n]<\infty$2, which enumerate the number of ways to partition sets subject to constraints, are: $E[|Y|^n]<\infty$3 and similarly for the associated Stirling numbers: $E[|Y|^n]<\infty$4 This representation positions the probabilistic heterogeneous Bell polynomials as generating functions for weighted set partitioning structures, parameterized via the moments of $E[|Y|^n]<\infty$5.

6. Special Case: Poisson and Bernoulli Distributions

For $E[|Y|^n]<\infty$6: $E[|Y|^n]<\infty$7 so

$E[|Y|^n]<\infty$8

with $E[|Y|^n]<\infty$9 classical Bell polynomials.

For $n \ge 0$0: $n \ge 0$1 yielding

$n \ge 0$2

indicating that the probabilistic polynomials collapse to the deterministic heterogeneous Bell polynomials with rescaled argument.

7. Interpretive Framework and Interpolative Role

The probabilistic heterogeneous Bell polynomials $n \ge 0$3 unify and interpolate among classical Bell, Lah–Bell, and probabilistic Bell polynomials, determined by $n \ge 0$4 and the distribution of $n \ge 0$5. They admit a moment-theoretic interpretation: for Poisson random variables $n \ge 0$6, the heterogeneous Bell polynomial $n \ge 0$7 equals the $n \ge 0$8th degenerate falling-factorial moment of $n \ge 0$9, and the analogy holds for the probabilistic version via the moments of $\lim_{n\to\infty} \frac{|t|^n E[|Y|^n]}{n!}=0$0. The entire structure recovers classical cases as $\lim_{n\to\infty} \frac{|t|^n E[|Y|^n]}{n!}=0$1 and encodes Lah-type combinatorics for $\lim_{n\to\infty} \frac{|t|^n E[|Y|^n]}{n!}=0$2 (Kim et al., 30 Mar 2025).

A plausible implication is that this formalism is extensible to other discrete probability models via appropriate choices of $\lim_{n\to\infty} \frac{|t|^n E[|Y|^n]}{n!}=0$3 and generalizations of the moment sequence. This framework produces a broad class of combinatorial quantities indexed by partitioning scheme, moment distribution, and heterogeneity parameter, positioning probabilistic heterogeneous Bell polynomials as a powerful organizing principle for enumeration and analysis in analytic combinatorics.