Stirling polynomials

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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[edit]

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) \,


Special values include:

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[edit]

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).

See also[edit]


  • Erdeli, A., Magnus, W. and Oberhettinger, F and Tricomi, F. G. Higher Transcendental Functions. Volume III:. New York. 

External links[edit]