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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

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 $$S_k(x)= B_k^{(x+1)}(x+1).$$

Bernoulli polynomials
Difference polynomials

References

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