# 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

${\displaystyle \left({t \over {1-e^{-t}}}\right)^{x+1}=\sum _{k=0}^{\infty }S_{k}(x){t^{k} \over k!}.}$

The first 10 Stirling polynomials are:

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

## Properties

Special values include:

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

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

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

{\displaystyle {\begin{aligned}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{aligned}}}

Here, ${\displaystyle L_{n}^{(\alpha )}}$ are Laguerre polynomials.

These following relations hold as well:

${\displaystyle {k+m \choose k}S_{k}(x-m)=\sum _{i=0}^{k}(-1)^{k-i}{k+m \choose i}S_{k-i+m,m}\cdot S_{i}(x),}$

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

${\displaystyle {k-m \choose k}S_{k}(x+m)=\sum _{i=0}^{k}{k-m \choose i}s_{m,m-k+i}\cdot S_{i}(x),}$

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

By differentiating the generating function it readily follows that

${\displaystyle S_{k}^{\prime }(x)=-\sum _{j=0}^{k-1}{k \choose j}S_{j}(x){\frac {B_{k-j}}{k-j}}.}$

## Relations to other polynomials

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

${\displaystyle \left({t \over {e^{t}-1}}\right)^{a}e^{zt}=\sum _{k=0}^{\infty }B_{k}^{(a)}(z){t^{k} \over k!}.}$

The relation is given by ${\displaystyle S_{k}(x)=B_{k}^{(x+1)}(x+1)}$.