---
title: Degenerate Stirling Numbers of the Second Kind
url: https://www.emergentmind.com/topics/degenerate-stirling-numbers-of-the-second-kind
type: topic
---

# Degenerate Stirling Numbers of the Second Kind

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 $S_2(n,k)$ [2201.07431][2205.01928][2410.12550][2305.04302][2412.03119][2501.05696].

### 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 $n > 0$ [2201.07431][2204.02595][2205.01928][2206.04402][2501.05696].

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 $(r)_{m,\lambda} = r(r-\lambda) \cdots (r-(m-1)\lambda)$ [2201.07431][2202.08421][1706.09681][2305.04302].

## 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 [2201.07431][2204.02595][2205.01928].

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 [2205.01928][2410.12550].

## 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 $\exp(x(e_{\lambda}(t)-1))$ [2201.07431][2204.02595][2205.01928][2202.04819][1706.09681].
- **Degenerate Fubini polynomials**: $F_{n,\lambda}(x) = \sum_{k=0}^n S_{2,\lambda}(n,k) k!\, x^k$ [2205.01928][2509.05761].
- **Degenerate Bernoulli polynomials**: $B_{n,\lambda}(x)$ with explicit expansions in terms of $S_{2,\lambda}(n,k)$ [2202.04819][2501.05696][2205.01928].
- **Degenerate Euler polynomials and numbers**: explicit identities and expansions with $S_{2,\lambda}(n,k)$ as coefficients [2205.01928][2412.03119].
- **Degenerate hyperharmonic numbers**: linear sums involving $S_{2,\lambda}(n,k)$ [2205.01928].
- **Degenerate Bell numbers and Dobinski-type formulas**: sums of the form
  \[
  B_{k,\lambda} = e^{-x} \sum_{m=0}^\infty \frac{(m)_{k,\lambda}}{m!} x^m
  \]
  [2204.02595][2202.04819][1706.09681][2501.05696].

## 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
\[
(a^\dagger a)_{k,\lambda} = \sum_{l=0}^k S_{2,\lambda}(k,l) (a^\dagger)^l a^l
\]
[2204.02595][2204.01252][2305.04302][2207.09997].

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 [2207.09997][1706.09681].

## 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 [2201.07431][2205.01928][1706.09681][2206.04402][2501.05696].

## 8. Limiting Behavior, Generalizations, and Open Problems

- In the limit $\lambda \to 0$, all degenerate objects revert to their classical counterparts: $(x)_{n,\lambda} \to x^n$, $e_{\lambda}(t) \to e^t$, $S_{2,\lambda}(n,k) \to S_2(n,k)$ [2201.07431][2205.01928][2410.12550][2206.04402][2412.03119][2501.05696].
- 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 $B(z)$ [2410.12550][2202.08421][2305.04302][1706.09681].
- No full combinatorial model is available for all $\lambda$, though weighted set partition and umbral probabilistic interpretations are suggested [2201.07431][2205.01928][2206.04402][2410.12550].
- 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 [2201.07431], [2204.02595], [2205.01928], [2204.01252], [2410.12550], [1706.09681], [2202.04819], [2206.04402], [2202.08421], [2305.04302], [2509.05761], [2412.03119], [2501.05696].

Source: https://www.emergentmind.com/topics/degenerate-stirling-numbers-of-the-second-kind