# Stirling polynomials

(Redirected from Stirling polynomial)

In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, the Stirling numbers and the Bernoulli numbers.

## Definition and examples

For nonnegative integers k, the Stirling polynomials Sk(x) are defined by the generating function equation

$\left( {t \over {1-e^{-t}}} \right) ^{x+1}= \sum_{k=0}^\infty S_k(x){t^k \over k!}.$

The first 10 Stirling polynomials are:

k $S_k(x)\,$
0 $1\,$
1 ${\scriptstyle\frac{1}{2}}(x+1)\,$
2 ${\scriptstyle\frac{1}{12}} (3x^2+5x+2) \,$
3 ${\scriptstyle\frac{1}{8}} (x^3+2x^2+x) \,$
4 ${\scriptstyle\frac{1}{240}} (15x^4+30x^3+5x^2-18x-8) \,$
5 ${\scriptstyle\frac{1}{96}} (3x^5+5x^4-5x^3-13x^2-6x) \,$
6 ${\scriptstyle\frac{1}{4032}} (63x^6+63x^5-315x^4-539x^3-84x^2+236x+96) \,$
7 ${\scriptstyle\frac{1}{1152}} (9x^7-84x^5-98x^4+91x^3+194x^2+80x) \,$
8 ${\scriptstyle\frac{1}{34560}} (135x^8-180x^7-1890x^6-840x^5+6055x^4+8140x^3+884x^2-3088x-1152) \,$
9 ${\scriptstyle\frac{1}{7680}} (15x^9-45x^8-270x^7+182x^6+1687x^5+1395x^4-1576x^3-2684x^2-1008x) \,$

## Properties

Special values include:

• $S_k(-m)= {(-1)^k \over {k+m-1 \choose k}} S_{k+m-1,m-1}$, where $S_{m,n}$ denotes Stirling numbers of the second kind. Conversely, $S_{n,m}=(-1)^{n-m} {n \choose m} S_{n-m}(-m-1)$;
• $S_k(-1)= \delta_{k,0};$
• $S_k(0)= (-1)^k B_k$, where $B_k$ are Bernoulli numbers;
• $S_k(1)= (-1)^{k+1} ((k-1) B_k+ k B_{k-1})$;
• $S_k(2)= {(-1)^{k}\over 2} ((k-1)(k-2) B_k+ 3 k(k-2) B_{k-1}+ 2 k(k-1) B_{k-2})$;
• $S_k(k)= k!$;
• $S_k(m)= {(-1)^k \over {m \choose k}} s_{m+1, m+1-k}$, where $s_{m,n}$ are Stirling numbers of the first kind. They may be recovered by $s_{n,m}= (-1)^{n-m} {n-1 \choose n-m} S_{n-m}(n-1)$.

The sequence $S_k(x-1)$ is of binomial type, since $S_k(x+y-1)= \sum_{i=0}^k {k \choose i} S_i(x-1) S_{k-i}(y-1)$. Moreover, this basic recursion holds: $S_k(x)= (x-k) {S_k(x-1) \over x} + k S_{k-1}(x+1)$.

Explicit representations involving Stirling numbers can be deduced with Lagrange's interpolation formula:

\begin{align}S_k(x)&= \sum_{n=0}^k (-1)^{k-n} S_{k+n,n} {{x+n \choose n} {x+k+1 \choose k-n} \over {k+n \choose n}} \\ &= \sum_{n=0}^k (-1)^n s_{k+n+1,n+1} {{x-k \choose n} {x-k-n-1 \choose k-n} \over {k+n \choose k}}\\ &= k! \sum_{j=0}^k (-1)^{k-j}\sum_{m=j}^k {x+m\choose m}{m\choose j}L_{k+m}^{(-k-j)}(-j).\end{align}

Here, $L_n^{(\alpha)}$ are Laguerre polynomials.

These following relations hold as well:

${k+m \choose k} S_k(x-m)= \sum_{i=0}^k (-1)^{k-i} {k+m \choose i} S_{k-i+m,m} \cdot S_i(x),$

where $S_{k,n}$ is the Stirling number of the second kind and

${k-m \choose k} S_k(x+m)= \sum_{i=0}^k {k-m \choose i} s_{m,m-k+i} \cdot S_i(x),$

where $s_{k,n}$ is the Stirling number of the first kind.

By differentiating the generating function it readily follows that

$S_k^\prime(x)=-\sum_{j=0}^{k-1} {k\choose j} S_j(x) \frac{B_{k-j}}{k-j}.$

## Relations to other polynomials

Closely related to Stirling polynomials are Nørlund polynomials (or generalized Bernoulli polynomials) with generating function

$\left( {t \over {e^t-1}} \right) ^a e^{z t}= \sum_{k=0}^\infty B^{(a)}_k(z){t^k \over k!}.$

The relation is given by $S_k(x)= B_k^{(x+1)}(x+1)$.