In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, which are closely related to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials. There are multiple variants of the Stirling polynomial sequence considered below most notably including the Sheffer sequence form of the sequence, ${\displaystyle S_{k}(x)}$, defined characteristically through the special form of its exponential generating function, and the Stirling (convolution) polynomials, ${\displaystyle \sigma _{n}(x)}$, which also satisfy a characteristic ordinary generating function and that are of use in generalizing the Stirling numbers (of both kinds) to arbitrary complex-valued inputs. We consider the "convolution polynomial" variant of this sequence and its properties second in the last subsection of the article. Still other variants of the Stirling polynomials are studied in the supplementary links to the articles given in the references.

Definition and examples

For nonnegative integers k, the Stirling polynomials, S_k(x), are a Sheffer sequence for ${\displaystyle (g(t),{\bar {f))(t)):=\left(e^{-t},\log \left({\frac {t}{1-e^{-t))}\right)\right)}$ ^[1] defined by the exponential generating function

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

The Stirling polynomials are a special case of the Nørlund polynomials (or generalized Bernoulli polynomials) ^[2] each with exponential generating function

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

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

The first 10 Stirling polynomials are given in the following table:

k	Sk(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	${\displaystyle {\scriptstyle {\frac {1}{7680))}(15x^{9}-45x^{8}-270x^{7}+182x^{6}+1687x^{5}+1395x^{4}-1576x^{3}-2684x^{2}-1008x)}$

Yet another variant of the Stirling polynomials is considered in ^[3] (see also the subsection on Stirling convolution polynomials below). In particular, the article by I. Gessel and R. P. Stanley defines the modified Stirling polynomial sequences, ${\displaystyle f_{k}(n):=S(n+k,n)}$ and ${\displaystyle g_{k}(n):=c(n,n-k)}$ where ${\displaystyle c(n,k):=(-1)^{n-k}s(n,k)}$ are the unsigned Stirling numbers of the first kind, in terms of the two Stirling number triangles for non-negative integers ${\displaystyle n\geq 1,\ k\geq 0}$. For fixed ${\displaystyle k\geq 0}$, both ${\displaystyle f_{k}(n)}$ and ${\displaystyle g_{k}(n)}$ are polynomials of the input ${\displaystyle n\in \mathbb {Z} ^{+))$ each of degree ${\displaystyle 2k}$ and with leading coefficient given by the double factorial term ${\displaystyle (1\cdot 3\cdot 5\cdots (2k-1))/(2k)!}$.

Properties

Below ${\displaystyle B_{k}(x)}$ denote the Bernoulli polynomials and ${\displaystyle B_{k}=B_{k}(0)}$ the Bernoulli numbers under the convention ${\displaystyle B_{1}=B_{1}(0)=-{\tfrac {1}{2));}$ ${\displaystyle s_{m,n))$ denotes a Stirling number of the first kind; and ${\displaystyle S_{m,n))$ denotes Stirling numbers of the second kind.

• Special values:
  $${\displaystyle {\begin{aligned}S_{k}(-m)&={\frac {(-1)^{k)){k+m-1 \choose k))S_{k+m-1,m-1}&&0<m\in \mathbb {Z} \\[6pt]S_{k}(-1)&=\delta _{k,0}\\[6pt]S_{k}(0)&=(-1)^{k}B_{k}\\[6pt]S_{k}(1)&=(-1)^{k+1}((k-1)B_{k}+kB_{k-1})\\[6pt]S_{k}(2)&={\tfrac {(-1)^{k)){2))((k-1)(k-2)B_{k}+3k(k-2)B_{k-1}+2k(k-1)B_{k-2})\\[6pt]S_{k}(k)&=k!\\[6pt]\end{aligned))}$$

  
• If ${\displaystyle m\in \mathbb {N} }$ and ${\displaystyle m<k}$ then:
  $${\displaystyle S_{k}(m)={(m+1)}{\binom {k}{m+1))\sum _{j=0}^{m}(-1)^{m-j}s_{m+1,m+1-j}{\frac {B_{k-j)){k-j)).}$$

  
• If ${\displaystyle m\in \mathbb {N} }$ and ${\displaystyle m\geq k}$ then:^[4]
  $${\displaystyle S_{k}(m)=(-1)^{k}B_{k}^{(m+1)}(0),}$$

  
  and:
  $${\displaystyle S_{k}(m)={(-1)^{k} \over {m \choose k))s_{m+1,m+1-k}.}$$

  

• 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:
  $${\displaystyle {\begin{aligned}S_{k}(x)&=\sum _{n=0}^{k}(-1)^{k-n}S_{k+n,n}((x+n \choose n}{x+k+1 \choose k-n} \over {k+n \choose n))\\[6pt]&=\sum _{n=0}^{k}(-1)^{n}s_{k+n+1,n+1}((x-k \choose n}{x-k-n-1 \choose k-n} \over {k+n \choose k))\\[6pt]&=k!\sum _{j=0}^{k}(-1)^{k-j}\sum _{m=j}^{k}{x+m \choose m}{m \choose j}L_{k+m}^{(-k-j)}(-j)\\[6pt]\end{aligned))}$$

  
  Here, ${\displaystyle L_{n}^{(\alpha )))$ are Laguerre polynomials.
• The following relations hold as well:
  $${\displaystyle {k+m \choose k}S_{k}(x-m)=\sum _{i=0}^{k}(-1)^{k-i}{k+m \choose i}S_{k-i+m,m}\cdot S_{i}(x),}$$

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

  
• By differentiating the generating function it readily follows that
  $${\displaystyle S_{k}^{\prime }(x)=-\sum _{j=0}^{k-1}{k \choose j}S_{j}(x){\frac {B_{k-j)){k-j)).}$$

  

Stirling convolution polynomials

Definition and examples

Another variant of the Stirling polynomial sequence corresponds to a special case of the convolution polynomials studied by Knuth's article ^[5] and in the Concrete Mathematics reference. We first define these polynomials through the Stirling numbers of the first kind as

${\displaystyle \sigma _{n}(x)=\left[{\begin{matrix}x\\x-n\end{matrix))\right]\cdot {\frac {1}{x(x-1)\cdots (x-n))).}$

It follows that these polynomials satisfy the next recurrence relation given by

${\displaystyle (x+1)\sigma _{n}(x+1)=(x-n)\sigma _{n}(x)+x\sigma _{n-1}(x),\ n\geq 1.}$

These Stirling "convolution" polynomials may be used to define the Stirling numbers, ${\displaystyle \scriptstyle {\left[{\begin{matrix}x\\x-n\end{matrix))\right]))$ and ${\displaystyle \scriptstyle {\left\((\begin{matrix}x\\x-n\end{matrix))\right\))}$, for integers ${\displaystyle n\geq 0}$ and arbitrary complex values of ${\displaystyle x}$. The next table provides several special cases of these Stirling polynomials for the first few ${\displaystyle n\geq 0}$.

n	σn(x)
0	${\displaystyle {\frac {1}{x))}$
1	${\displaystyle {\frac {1}{2))}$
2	${\displaystyle {\frac {1}{24))(3x-1)}$
3	${\displaystyle {\frac {1}{48))(x^{2}-x)}$
4	${\displaystyle {\frac {1}{5760))(15x^{3}-30x^{2}+5x+2)}$
5	${\displaystyle {\frac {1}{11520))(3x^{4}-10x^{3}+5x^{2}+2x)}$
6	${\displaystyle {\frac {1}{2903040))(63x^{5}-315x^{4}+315x^{3}+91x^{2}-42x-16)}$
7	${\displaystyle {\frac {1}{5806080))(9x^{6}-63x^{5}+105x^{4}+7x^{3}-42x^{2}-16x)}$
8	${\displaystyle {\frac {1}{1393459200))(135x^{7}-1260x^{6}+3150x^{5}-840x^{4}-2345x^{3}-540x^{2}+404x+144)}$
9	${\displaystyle {\frac {1}{2786918400))(15x^{8}-180x^{7}+630x^{6}-448x^{5}-665x^{4}+100x^{3}+404x^{2}+144x)}$
10	${\displaystyle {\frac {1}{367873228800))(99x^{9}-1485x^{8}+6930x^{7}-8778x^{6}-8085x^{5}+8195x^{4}+11792x^{3}+2068x^{2}-2288x-768)}$

Generating functions

This variant of the Stirling polynomial sequence has particularly nice ordinary generating functions of the following forms:

${\displaystyle {\begin{aligned}\left({\frac {ze^{z)){e^{z}-1))\right)^{x}&=\sum _{n\geq 0}x\sigma _{n}(x)z^{n}\\\left({\frac {1}{z))\ln {\frac {1}{1-z))\right)^{x}&=\sum _{n\geq 0}x\sigma _{n}(x+n)z^{n}.\end{aligned))}$

More generally, if ${\displaystyle {\mathcal {S))_{t}(z)}$ is a power series that satisfies ${\displaystyle \ln \left(1-z{\mathcal {S))_{t}(z)^{t-1}\right)=-z{\mathcal {S))_{t}(z)^{t))$, we have that

${\displaystyle {\mathcal {S))_{t}(z)^{x}=\sum _{n\geq 0}x\sigma _{n}(x+tn)z^{n}.}$

We also have the related series identity ^[6]

${\displaystyle \sum _{n\geq 0}(-1)^{n-1}\sigma _{n}(n-1)z^{n}={\frac {z}{\ln(1+z)))=1+{\frac {z}{2))-{\frac {z^{2)){12))+\cdots ,}$

and the Stirling (Sheffer) polynomial related generating functions given by

${\displaystyle \sum _{n\geq 0}(-1)^{n+1}m\cdot \sigma _{n}(n-m)z^{n}=\left({\frac {z}{\ln(1+z)))\right)^{m))$

${\displaystyle \sum _{n\geq 0}(-1)^{n+1}m\cdot \sigma _{n}(m)z^{n}=\left({\frac {z}{1-e^{-z))}\right)^{m}.}$

Properties and relations

For integers ${\displaystyle 0\leq k\leq n}$ and ${\displaystyle r,s\in \mathbb {C} }$, these polynomials satisfy the two Stirling convolution formulas given by

${\displaystyle (r+s)\sigma _{n}(r+s+tn)=rs\sum _{k=0}^{n}\sigma _{k}(r+tk)\sigma _{n-k}(s+t(n-k))}$

and

${\displaystyle n\sigma _{n}(r+s+tn)=s\sum _{k=0}^{n}k\sigma _{k}(r+tk)\sigma _{n-k}(s+t(n-k)).}$

When ${\displaystyle n,m\in \mathbb {N} }$, we also have that the polynomials, ${\displaystyle \sigma _{n}(m)}$, are defined through their relations to the Stirling numbers

${\displaystyle {\begin{aligned}\left\((\begin{matrix}n\\m\end{matrix))\right\}&=(-1)^{n-m+1}{\frac {n!}{(m-1)!))\sigma _{n-m}(-m)\ ({\text{when ))m<0)\\\left[{\begin{matrix}n\\m\end{matrix))\right]&={\frac {n!}{(m-1)!))\sigma _{n-m}(n)\ ({\text{when ))m>n),\end{aligned))}$

and their relations to the Bernoulli numbers given by

${\displaystyle {\begin{aligned}\sigma _{n}(m)&={\frac {(-1)^{m+n-1)){m!(n-m)!))\sum _{0\leq k<m}\left[{\begin{matrix}m\\m-k\end{matrix))\right]{\frac {B_{n-k)){n-k)),\ n\geq m>0\\\sigma _{n}(m)&=-{\frac {B_{n)){n\cdot n!)),\ m=0.\end{aligned))}$