---
title: 'p-Stirling Numbers: Concepts & Applications'
url: https://www.emergentmind.com/topics/p-stirling-numbers
type: topic
---

# p-Stirling Numbers: Concepts & Applications

A $p$-Stirling number is a generalization of the classical Stirling numbers of both the first and second kind, parameterized by an integer $p$ and possessing deep combinatorial, algebraic, and $p$-adic properties. These numbers admit distinct formal definitions depending on context, appearing in symmetric function theory, operator calculus, polynomial expansions, and $p$-adic arithmetic. Below is a technical survey of their principal definitions, recurrence relations, generating functions, combinatorial models, $p$-adic behavior, and applications.

## 1. Definitions and Fundamental Properties

The $p$-Stirling numbers have two principal interpretations in the literature:

- **(a) Classical $p$-Stirling numbers (parameter extension):**  
  As defined by Merris, Sun, and formalized in the context of symmetric functions [1302.4694], the $p$-Stirling numbers of the first and second kind, denoted $c_p(n,k)$ and $S_p(n,k)$, are given via symmetric function operations:
  $$
  c_p(n,k) = e_{n-k}(p+n-1,p+n-2,\ldots,p), \quad 
  S_p(n,k) = h_{n-k}(p+k,p+k-1,\ldots,p),
  $$
  where $e_{r}(\cdot)$ and $h_r(\cdot)$ denote the elementary and complete homogeneous symmetric functions, respectively.

  These numbers satisfy the recursions:
  $$
  \begin{align*}
  c_p(n,k) &= c_p(n-1,k-1) + (p+n-1) c_p(n-1,k), \\
  S_p(n,k) &= S_p(n-1,k-1) + (p+k) S_p(n-1,k),
  \end{align*}
  $$
  with boundary conditions $c_p(0,0)=S_p(0,0)=1$, $c_p(n,0)=\prod_{j=0}^{n-1}(p+j)$, and $S_p(n,n)=1$ [1302.4694].

  **Explicit formula**:
  $$
  S_p(n,k) = \frac{1}{k!} \sum_{j=0}^k (-1)^{k-j} \binom{k}{j} (p+j)^n
  $$
  [1302.4694].

- **(b) Generalized (Parameter-Embedded) Stirling Numbers:**  
  In the context of polynomial expansions and operator calculus, the term "$p$-Stirling number" can also refer to the coefficients in the expansion of $(a n+b)^p$ (or more generally, products involving such powers) into the basis of falling factorials $(n)_k$:
  $$
  (a n+b)^p = \sum_{k=0}^p k! \, S(p,k) (n)_k
  $$
  where $S(p,k)$ is the classical $p$-Stirling number of the second kind ($a=1$, $b=0$) [1803.05953]. The generalization allows arbitrary $a$, $b$, and higher multiplicities.

  The explicit formula in this interpretation is:
  $$
  S(p,k) = \frac{1}{k!} \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} j^p
  $$
  [1803.05953].

## 2. Generating Functions and Algebraic Structure

Both variants possess closed-form generating functions and inversion relations:

| Type                 | Generating Function                                                                            | Inverse Relation                            |
|----------------------|-----------------------------------------------------------------------------------------------|---------------------------------------------|
| $S_p(n,k)$           | $\displaystyle\sum_{n \ge k} S_p(n,k) x^n = \frac{x^k}{\prod_{j=0}^k (1-(p+j)x)}$              | $\sum_{j=m}^n (-1)^{n-j} c_p(n,j) S_p(j,m) = \delta_{n,m}$        |
| $c_p(n,k)$           | $\prod_{j=0}^{n-1} (x + (p+j)) = \sum_{k=0}^n c_p(n,k) x^k$                   | $[c_p(n,k)]$ and $[(-1)^{n-k} S_p(n,k)]$ inverse lower-triangular |

Explicit formula for $S_p(n,k)$ above generalizes the Dobinski formula for ordinary Stirling numbers [1302.4694][1803.05953].

## 3. Combinatorial and Algebraic Interpretations

- **Symmetric Function Model:** The definitions via $e_{r}$ and $h_{r}$ relate $p$-Stirling numbers to evaluations of symmetric functions at shifted integer sequences, connecting to the ring of symmetric polynomials [1302.4694].

- **Colored Permutations and Set Partitions:** $c_p(n,k)$ counts permutations of $[n]+\{n+1,\ldots,n+p\}$ into $k+p$ disjoint cycles, where the distinguished $p$ elements occur in separate cycles; $S_p(n,k)$ counts partitions of $[n]+\{n+1,\ldots,n+p\}$ into $k+p$ blocks, each special block containing one of the distinguished $p$ elements [1302.4694].

- **Operator Calculus and Forest Models:** In the context of operator expansions, e.g., $(x^k D)^p$, the coefficients generalize Stirling numbers with a bivariate $(p,k)$-parameter and admit combinatorial interpretations via $(p,k)$-forests and colored increasing trees [1012.3948].

## 4. $p$-adic Properties and Asymptotic Behavior

The $p$-Stirling numbers (classical or generalized) exhibit rich $p$-adic behavior, notably:

- **$p$-adic Valuations of Stirling Numbers:** Explicit $p$-adic valuation formulas exist for both $s(n,k)$ and $S(n,k)$ in certain congruence classes, such as $n = a p^h$, $k = b p^h + a$ [1805.00995][2111.08766]. For instance,
  $$
  \nu_p(S(n,k)) = \left\lceil \frac{\sigma_p(k) - \sigma_p(n)}{p-1} \right\rceil,
  $$
  where $\sigma_p(m)$ denotes the sum of the base-$p$ digits [1805.00995, Lemma 2.1]. In the case $p=2$,
  $$
  \nu_2(S(2^h, k)) = d_2(k) - 1,
  $$
  where $d_2(k)$ is the number of $1$'s in the binary expansion of $k$ [1805.00995].

- **$p$-adic Analytic Continuation:** For fixed $k$, the sequence $(S(n,k))_{n \in \mathbb{N}}$ admits a locally analytic $p$-adic extension, with nontrivial congruence patterns arising from decomposition into $p$-adic "balls" and controlled by the residue class of $n$ modulo powers of $p$ and $p-1$ [1803.04533].

- **$p$-adic Limits and Infinite Arrays:** For suitable progressions in $n$ and $k$, $(S(p^e a+c, p^e b+d))_{e\ge 0}$ has a $p$-adic limit, defining $S(p^\infty a+c, p^\infty b+d) \in \mathbb{Z}_p$, with explicit closed forms when $a \equiv b \pmod{p-1}$ [1307.7687].

## 5. Generalizations and Applications

- **Generalized Stirling Numbers:** The $p$-Stirling numbers fall within a much wider class of "generalized Stirling numbers" arising from the expansion of general polynomials $\prod_s (a_s n + b_s)^{p_s}$ into the falling factorial basis, with explicit recurrences and Dobinski-type formulas [1803.05953]. Such numbers interpolate between the classical cases, $p$-Stirling numbers, and numerous other objects (e.g., degenerate, Lah, central factorial numbers) [2202.11306].

- **Convolution Relations, Orthogonality, and Determinants:** There exist orthogonality relations:
  $$
  \sum_{j=m}^n (-1)^{n-j} c_p(n,j) S_p(j,m) = \delta_{n,m},
  $$
  convolution formulas relating parameters $p_1$, $p_2$ (e.g., $S_{p_1+p_2}(n,k)$ in terms of $S_{p_1}$ and $S_{p_2}$), and determinantal expressions for block matrices built from $c_p(n,k)$ [1302.4694].

## 6. Connections with Polynomial Sequences and Umbral Calculus

The study of $p$-Stirling numbers is embedded in the wider project of associating generalized Stirling numbers with sequences of polynomials $P = \{p_n(x)\}$, where the expansion
$$
p_n(x) = \sum_{k=0}^n S_P(n,k) (x)_k
$$
defines "P–Stirling numbers of the second kind" and a corresponding family of first-kind numbers inverts this relationship. Specializations of $P$ recover $p$-Stirling numbers, central factorial numbers, Bernoulli, Euler, and Lah numbers, and others [2202.11306].

Explicit generating functions, recurrences, and matrix inversion identities apply uniformly in the Sheffer–umbral framework:
$$
\sum_{n \ge k} S_P(n,k) \frac{t^n}{n!} = \frac{1}{k!} [g(t)(e^{f(t)}-1)]^k,
$$
where $(g,f)$ are Sheffer parameters for the sequence $P$ [2202.11306].

## 7. Research Directions and Open Problems

Current research trends include:

- Characterization of $p$-adic valuation patterns and their stabilization, including fine-scale decompositions of $p$-adic "balls" for Stirling number sequences [1803.04533][2111.08766].
- Systematic investigation of $p$-adic limits of combinatorial number arrays and the construction of analytic interpolants [1307.7687][1402.0433].
- Elucidation of combinatorial interpretations for generalized and weighted Stirling numbers, especially via forest and colored permutation models [1012.3948][1302.4694].
- The role of higher-order Bernoulli numbers and their pole structure in determining $p$-adic behavior [1805.00995].

A notable direction is the complete characterization of $p$-adic valuation jumps in towers of the first-kind Stirling numbers, as partially confirmed for odd $k$ by uniform "slope 2" phenomena [1908.05594], and the continued extension of umbral and operator methods to uncover new families of Stirling-like numbers.

---

**References**:  
- [1302.4694]  
- [1012.3948]  
- [1803.05953]  
- [2202.11306]  
- [1805.00995]  
- [2111.08766]  
- [1908.05594]  
- [1803.04533]  
- [1307.7687]  
- [1402.0433]

Source: https://www.emergentmind.com/topics/p-stirling-numbers