Degenerate Stirling Numbers of the Second Kind

• Degenerate Stirling numbers of the second kind are defined as the coefficients in the expansion of the degenerate falling factorial, generalizing classical partitions using the deformation parameter λ.
• They feature explicit generating functions, recurrences, and closed-form expressions that bridge algebraic formulations with analytic and combinatorial applications.
• These numbers are pivotal in fields such as degenerate combinatorics, quantum operator theory, and bosonic normal ordering, offering insights into weighted set partitions and polynomial systems.

The degenerate Stirling numbers of the second kind, denoted $S_{2,\lambda}(n,k)$, generalize the classical Stirling numbers by introducing a deformation parameter $\lambda$. These numbers enumerate, with a nontrivial weighting structure, partitions of an $n$-element set into $k$ blocks, and appear frequently in the analysis of deformed polynomial systems, degenerate versions of special functions, and bosonic normal-ordering problems. Their algebraic and analytic properties are governed by $\lambda$, interpolating smoothly to classical results as $\lambda \to 0$, and they form the backbone of numerous degenerate combinatorial and operator-theoretic identities.

1. Algebraic Definition and Expansion

Let $\lambda \in \mathbb{R}$. The degenerate falling factorial is

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

The degenerate Stirling numbers of the second kind are the coefficients in the expansion

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

or equivalently, the inverse expansion

$$x^n = \sum_{k=0}^n S_{2,\lambda}(n,k)\, (x)_{k,\lambda} [2201.07431][2204.02595][2205.01928][2410.12550].$$

They reduce to the classical Stirling numbers: $$\lim_{\lambda \to 0} S_{2,\lambda}(n,k) = S_2(n,k).$$

Combinatorially, $S_{2,\lambda}(n,k)$ enumerates weighted set partitions, where each block of size $m$ contributes a factor

$\prod_{j=0}^{m-1}(1-j \lambda),$

recovering $S_2(n,k)$ when $\lambda = 0$.

2. Generating Functions

Exponential Generating Function

The degenerate exponential function is

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

The exponential generating function for fixed $k$ is

$$\frac{(e_{\lambda}(t) - 1)^k}{k!} = \sum_{n=k}^\infty S_{2,\lambda}(n,k) \frac{t^n}{n!} [2201.07431][2204.02595][2205.01928].$$

The bivariate exponential generating function is

$$\exp(x(e_{\lambda}(t) - 1)) = \sum_{n=0}^\infty (x)_{n,\lambda} \frac{t^n}{n!} = \sum_{k=0}^\infty x^k \sum_{n=k}^\infty S_{2,\lambda}(n,k)\frac{t^n}{n!}.$$

Ordinary Generating Function

For $n$ fixed,

$$\sum_{k=0}^n S_{2,\lambda}(n,k)\, u^k = (u)(u + \lambda)\cdots(u + (n-1)\lambda) [2204.02595][2501.05696].$$

3. Closed-Form Expressions and Recurrences

Inclusion–Exclusion/Explicit Formula

The principal explicit formula is

$$S_{2,\lambda}(n,k) = \frac{1}{k!} \sum_{j=0}^k (-1)^{k-j} \binom{k}{j} (j)_{n,\lambda},$$

which generalizes the classical inclusion–exclusion formula for $S_2(n,k)$ (Kim et al., 2022, Kim et al., 2022, Adell et al., 2024, Kim et al., 2023, Kim et al., 2024, Kim et al., 10 Jan 2025).

Fundamental Recurrence

The numbers satisfy a triangular recurrence

$$S_{2,\lambda}(n+1,k) = S_{2,\lambda}(n,k-1) + (k - n\lambda)\, S_{2,\lambda}(n,k), \qquad n \ge 0,\, 1 \le k \le n+1,$$

with boundary $S_{2,\lambda}(0,0) = 1$, $S_{2,\lambda}(n,0) = 0$ for $n > 0$ (Kim et al., 2022, Kim et al., 2022, Kim et al., 2022, Kim et al., 2022, Kim et al., 10 Jan 2025).

Alternative forms appear as

$$S_{2,\lambda}(n,k) = S_{2,\lambda}(n-1,k-1) + (k - (n-1)\lambda) S_{2,\lambda}(n-1,k),$$

or, in a shifted notation,

$$S_{2,\lambda}(n+1,k) = S_{2,\lambda}(n,k-1) - n\lambda S_{2,\lambda}(n,k) [2206.04402].$$

Higher-Order and r-Shifted Degenerate Stirling Numbers

The degenerate $r$-Stirling numbers of the second kind are defined by

$$(e_{\lambda}(t)-1)^k\, e_{\lambda}(r t) = k! \sum_{n=k}^\infty S_{2,\lambda}^{(r)}(n+r, k+r) \frac{t^n}{n!},$$

with

$$S_{2,\lambda}^{(r)}(n+r, k+r) = \sum_{\ell=k}^n S_{2,\lambda}(\ell, k) (r)_{n-\ell, \lambda},$$

where $$(r)_{m,\lambda} = r(r-\lambda) \cdots (r-(m-1)\lambda)$$ (Kim et al., 2022, Kim et al., 2022, Kim et al., 2017, Kim et al., 2023).

4. Orthogonality, Inversion, and Umbral Structure

The degenerate Stirling numbers of the second kind invert those of the first kind. Let $S_{1,\lambda}(n,k)$ be defined by

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

then

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

and analogously for the reverse sum (Kim et al., 2022, Kim et al., 2022, Kim et al., 2022).

These relations endow the arrays $\{S_{2,\lambda}(n,k)\}$ and $\{S_{1,\lambda}(n,k)\}$ with a matrix-inverse structure, facilitating basis changes in polynomial expansions, and linking to umbral calculus and probabilistic cumulant–moment relationships (Kim et al., 2022, Adell et al., 2024).

5. Interplay with Degenerate Polynomials and Special Numbers

The degenerate Stirling numbers of the second kind underpin numerous constructions in degenerate combinatorics, including:

• Degenerate Bell polynomials: $$\Phi_{n,\lambda}(x) = \sum_{k=0}^n S_{2,\lambda}(n,k) x^k$$, with generating function $\exp(x(e_{\lambda}(t)-1))$ (Kim et al., 2022, Kim et al., 2022, Kim et al., 2022, Kim et al., 2022, Kim et al., 2017).
• Degenerate Fubini polynomials: $$F_{n,\lambda}(x) = \sum_{k=0}^n S_{2,\lambda}(n,k) k!\, x^k$$ (Kim et al., 2022, Kim et al., 6 Sep 2025).
• Degenerate Bernoulli polynomials: $B_{n,\lambda}(x)$ with explicit expansions in terms of $S_{2,\lambda}(n,k)$ (Kim et al., 2022, Kim et al., 10 Jan 2025, Kim et al., 2022).
• Degenerate Euler polynomials and numbers: explicit identities and expansions with $S_{2,\lambda}(n,k)$ as coefficients (Kim et al., 2022, Kim et al., 2024).
• Degenerate hyperharmonic numbers: linear sums involving $S_{2,\lambda}(n,k)$ (Kim et al., 2022).
• Degenerate Bell numbers and Dobinski-type formulas: sums of the form

$$B_{k,\lambda} = e^{-x} \sum_{m=0}^\infty \frac{(m)_{k,\lambda}}{m!} x^m$$

(Kim et al., 2022, Kim et al., 2022, Kim et al., 2017, Kim et al., 10 Jan 2025).

6. Applications in Operator Theory and Quantum Calculus

Degenerate Stirling numbers of the second kind naturally arise as structure coefficients in the normal ordering of powers of the number operator in boson algebra

$$(a^\dagger a)_{k,\lambda} = \sum_{l=0}^k S_{2,\lambda}(k,l) (a^\dagger)^l a^l$$

(Kim et al., 2022, Kim et al., 2022, Kim et al., 2023, Kim et al., 2022).

This operator-theoretic representation connects degenerate Stirling numbers to coherent-state expansions, nonclassical statistics, and generalizations of partition algebras in quantum analysis.

The degenerate $r$-Stirling numbers, being the coefficients in normal ordering $(a^\dagger a)_{m,\lambda} (a^\dagger)^r$, have analogous bosonic interpretations (Kim et al., 2022, Kim et al., 2017).

7. Computational Aspects and Explicit Tables

Practical computation proceeds via dynamic programming using the recurrence relations and closed-form formulas. Complexity is $O(n^2)$, and for small $n,k$, explicit tables can be constructed as follows:

$n\backslash k$	$0$	$1$	$2$	$3$
$0$	$1$	$0$	$0$	$0$
$1$	$0$	$1$	$0$	$0$
$2$	$0$	$1-\lambda$	$1$	$0$
$3$	$0$	$1-3\lambda+2\lambda^2$	$3-3\lambda$	$1$

As $\lambda \to 0$, these entries recover the standard Stirling triangle (Kim et al., 2022, Kim et al., 2022, Kim et al., 2017, Kim et al., 2022, Kim et al., 10 Jan 2025).

8. Limiting Behavior, Generalizations, and Open Problems

• In the limit $\lambda \to 0$, all degenerate objects revert to their classical counterparts: $(x)_{n,\lambda} \to x^n$, $e_{\lambda}(t) \to e^t$, $S_{2,\lambda}(n,k) \to S_2(n,k)$ (Kim et al., 2022, Kim et al., 2022, Adell et al., 2024, Kim et al., 2022, Kim et al., 2024, Kim et al., 10 Jan 2025).
• Degenerate Stirling numbers generalize naturally to $r$-Stirling and $(r,s)$-Stirling frameworks, and to the structure coefficients of $B$-Stirling numbers for general analytic $B(z)$ (Adell et al., 2024, Kim et al., 2022, Kim et al., 2023, Kim et al., 2017).
• No full combinatorial model is available for all $\lambda$, though weighted set partition and umbral probabilistic interpretations are suggested (Kim et al., 2022, Kim et al., 2022, Kim et al., 2022, Adell et al., 2024).
• Open directions include explicit combinatorial models of $\lambda$-weighted partitions, noncommutative generalizations, and connections to probabilistic and statistical mechanics constructions.

Bibliography

Principal references for all definitions, recurrence relations, generating functions, combinatorial interpretations, operator-theoretic applications, and closed-form formulas are (Kim et al., 2022, Kim et al., 2022, Kim et al., 2022, Kim et al., 2022, Adell et al., 2024, Kim et al., 2017, Kim et al., 2022, Kim et al., 2022, Kim et al., 2022, Kim et al., 2023, Kim et al., 6 Sep 2025, Kim et al., 2024, Kim et al., 10 Jan 2025).