Fully Degenerate Bell Polynomials

• Fully degenerate Bell polynomials are λ-deformed Bell numbers defined by replacing standard exponentials and factorials with their degenerate counterparts.
• They are generated via degenerate exponential functions and falling factorials, leading to explicit expansions and Spivey-type recurrences.
• These polynomials interpolate between classical set partitions and pure powers, providing essential tools in umbral calculus, combinatorics, and nonclassical probability.

A fully degenerate Bell polynomial is a one-parameter $\lambda$ deformation of the classical Bell polynomials, defined by replacing all standard exponentials and falling/rising factorials with their degenerate $\lambda$ analogues. This interpolation between Bell polynomials and powers $x^n$ is characterized by a rich algebraic and combinatorial structure, including connections to degenerate Stirling numbers, operator identities, Spivey-type recurrences, and degenerate Poisson-type moment representations. The paper of these polynomials provides essential tools for advances in umbral calculus, combinatorics, nonclassical probability, and operator theory.

1. Fundamental Definitions and Generating Functions

The fully degenerate Bell polynomials $B_{n,\lambda}(x)$ (many variants are present in the literature, but the archetypal version is discussed here) are indexed by integers $n\ge 0$ and parameterized by $\lambda\in\mathbb{R}$. They use the degenerate falling factorial

$$(x)_{n,\lambda} = x(x-\lambda)(x-2\lambda)\cdots(x-(n-1)\lambda), \quad (x)_{0,\lambda}=1,$$

and the degenerate exponential

$$e_\lambda^x(t) = \sum_{k=0}^\infty (x)_{k,\lambda}\frac{t^k}{k!} = (1+\lambda t)^{x/\lambda}, \quad e_\lambda(t)=e_\lambda^1(t).$$

The exponential generating function (EGF) is

$$\sum_{n=0}^\infty B_{n,\lambda}(x)\frac{t^n}{n!} = e_\lambda^x(e_\lambda(t)-1).$$

The special value $B_{n,\lambda} = B_{n,\lambda}(1)$ is called the fully degenerate Bell number. As $\lambda\to 0$, this reduces to the classical Bell (Touchard) polynomial generating function $\exp(x(e^t-1))$, so

$\lim_{\lambda\to0}B_{n,\lambda}(x) = B_n(x).$

This definition—using two degenerate exponentials—distinguishes the fully degenerate (or doubly degenerate) theory from the various partially or singly degenerate cases (Kim et al., 2021, Kim et al., 2021, Kim et al., 6 Sep 2025).

2. Explicit Expansions and Stirling Connections

The degenerate Stirling numbers of the second kind $S_{2,\lambda}(n,k)$ are defined through

$$x^n = \sum_{k=0}^n S_{2,\lambda}(n,k)\,(x)_{k,\lambda}.$$

The fully degenerate Bell polynomials can be expanded explicitly as

$$B_{n,\lambda}(x) = \sum_{k=0}^n S_{2,\lambda}(n,k)\,x^k.$$

Table 1 illustrates $B_{n,\lambda}(x)$ for small $n$:

$n$	$B_{n,\lambda}(x)$
0	$1$
1	$x$
2	$(1-\lambda)x + x^2$
3	$(1-\lambda)(1-2\lambda)x + 3(1-\lambda)x^2 + x^3$
4	$$(1-\lambda)(1-2\lambda)(1-3\lambda)x + 6(1-\lambda)(1-2\lambda)x^2$$<br>$+ 7(1-\lambda)x^3 + x^4$

The degeneration parameter $\lambda$ interpolates the combinatorics between classical set partitions ($\lambda=0$) and pure powers ($\lambda=1$): $B_{n,1}(x)=x^n$ (Kim et al., 2021, Ma et al., 2021).

3. Recurrence Relations and Operator Representations

Several recurrence relations generalize classical identities:

• Mixed falling-factorial recurrence:

$$B_{n+1,\lambda} = \sum_{m=0}^n \binom{n}{m}\,B_{m,\lambda}\,(1)_{n-m+1,\lambda}.$$

• Derivative-type recurrence:

$$B_{n+1,\lambda}(x) = x\bigl(B'_{n,\lambda}(x) + B_{n,\lambda}(x)\bigr) - n\lambda B_{n,\lambda}(x).$$

• Addition formula:

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

• Operator formula:

$$(x-\lambda D)^n e^{xt} = (x)_{n,\lambda} e^{xt}, \quad D = \frac{d}{dx}.$$

• Antiderivative via degenerate Bernoulli numbers:

$$\int_0^x B_{n,\lambda}(t)\,dt = \frac{1}{n+1} \sum_{k=0}^n \binom{n+1}{k} \beta_{n+1-k,\lambda} B_{k,\lambda}(x),$$

where $\beta_{n,\lambda}$ are Carlitz's degenerate Bernoulli numbers.

These recurrences encode both combinatorial and differential-algebraic structure and are foundational tools for theoretical and computational applications (Kim et al., 2021, Kim et al., 2022, Ma et al., 2021).

4. Spivey-Type Double Sum and Refined Recurrences

The Spivey-type recurrence provides a two-parameter generalization: $$B_{n+m,\lambda}(x) = \sum_{k=0}^m \sum_{l=0}^n (1)_{k,\lambda} {m \brace k}_{\lambda} \binom{n}{l} x^k F^{(k)}_{n-l,\lambda}(-\lambda x, k - m\lambda) B_{l,\lambda}(x),$$ where $F^{(k)}_{n,\lambda}(u, y)$ are two-variable degenerate Fubini polynomials (Kim et al., 6 Sep 2025). More recent operator-based approaches derive natural Spivey-type recurrences without introducing extraneous families such as two-variable Fubini polynomials, using operator algebras: $$B_{n+m,\lambda}(x) = \sum_{j=0}^m \sum_{k=0}^n {m\brace j}_{\lambda} \binom{n}{k} (j - m\lambda)_{n-k,\lambda} (1)_{j,\lambda} x^j B_{k,\lambda}(x, 1 - j\lambda),$$ with $B_{k,\lambda}(x, y)$ a two-variable generalization (Kim et al., 16 Nov 2025). Such recursions generalize the classical Spivey recurrence, allowing for computation and structural insight in degenerate regimes.

5. Degenerate Stirling Numbers: Orthogonality and Inversions

In the degenerate context, both first and second kind degenerate Stirling numbers arise, satisfying mutual orthogonality relations. Two fundamental inversion identities are

$$(x)_{n,\lambda} = \sum_{k=0}^n S_{1,\lambda}(n,k) x^k,\qquad x^n = \sum_{k=0}^n S_{2,\lambda}(n,k) (x)_{k,\lambda},$$

with the inversion formula

$$x^n = \sum_{k=0}^n (-1)^{n-k} S_{1,\lambda}(n,k) \sum_{j=0}^k S_{2,\lambda}(k,j) x^j.$$

These relations encode the interplay between the degenerate analogues of power and falling-factorial bases, and facilitate the derivation of numerous identities for connections between various degenerate polynomial sequences (Kim et al., 2021).

6. Combinatorial and Probabilistic Interpretations

For $\lambda=0$, $B_{n,\lambda}(x)$ counts set partitions with $x$ marked per block. In the fully degenerate regime, $B_{n,\lambda}(x)$ is the $n$th moment for a so-called "degenerate Poisson" law,

$$\mathbb{P}\{X = k\} = \frac{(x)_{k,\lambda}}{k!}\,e_\lambda(-x),$$

implying $B_{n,\lambda}(x) = \mathbb{E}[X^n]$ (Kim et al., 2021, Kim et al., 2021, Kim et al., 2022). Weighted set partition models arise, where each block's size-dependent weight is controlled by $\lambda$, leading to deformations of classical enumeration and moment theory.

7. Extensions, Special Cases, and Applications

Numerous extensions of the fully degenerate Bell polynomials have been introduced:

• $r$-Bell polynomials: Additional $r$-parameter shifts in the generating function, connected to degenerate bosonic normal ordering problems and appearing naturally in operator/algebraic combinatorics (Kim et al., 2022, Kim et al., 2023).
• Generalized $(r,s)$-Bell polynomials: Studied via generalized degenerate Stirling numbers, applicable in deformed algebraic structures and arising in quantum normal ordering (Kim et al., 2023).
• Sheffer-type and umbral calculus: The sequence $B_{n,\lambda}(x)$ forms a Sheffer sequence for a specific pair of generating function ingredients, allowing umbral and operator-theoretic identities and facilitating explicit computation, addition formulas, and lowering/rising operators (Ma et al., 2021, Kim et al., 2017).
• Probabilistic models: Fully degenerate Bell polynomials encode moments of degenerate Poisson random variables, giving nonclassical distributions.

The fully degenerate Bell polynomial framework assimilates classical, degenerate, and generalized combinatorial and algebraic objects, acting as a robust toolkit for both foundational mathematics and applied fields requiring deformation or interpolation between discrete structures (Kim et al., 2021, Kim et al., 2022, Kim et al., 6 Sep 2025, Kim et al., 16 Nov 2025).