Stirling polynomials

Definition and examples

For nonnegative integers k, the Stirling polynomials, S**k(x), are a Sheffer sequence for (g(t), \bar{f}(t)) := \left(e^{-t}, \log\left(\frac{t}{1-e^{-t}}\right)\right) defined by the exponential generating function ::\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) each with exponential 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!},

given by the relation S_k(x)= B_k^{(x+1)}(x+1).

The first 10 Stirling polynomials are given in the following table: :{| class="wikitable" !k !! Sk(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) |} Yet another variant of the Stirling polynomials is considered in (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, f_k(n) := S(n+k, n) and g_k(n) := c(n, n-k) where 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 n \geq 1,\ k \geq 0. For fixed k \geq 0, both f_k(n) and g_k(n) are polynomials of the input n \in \mathbb{Z}^{+} each of degree 2k and with leading coefficient given by the double factorial term (1 \cdot 3 \cdot 5 \cdots (2k-1)) / (2k)!.

Properties

Below B_k(x) denote the Bernoulli polynomials and B_k = B_k(0) the Bernoulli numbers under the convention B_1 = B_1(0) = -\tfrac{1}{2}; s_{m,n} denotes a Stirling number of the first kind; and S_{m,n} denotes Stirling numbers of the second kind.

• Special values: \begin{align} S_k(-m) &= \frac{(-1)^k} S_{k+m-1,m-1} && 0 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+ k B_{k-1}) \[6pt] S_k(2) &= \tfrac{(-1)^k}{2} ((k-1)(k-2) B_k+ 3 k(k-2) B_{k-1}+ 2 k(k-1) B_{k-2}) \[6pt] S_k(k) &= k! \[6pt] \end{align}

• If m \in \N and m then: 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 m \in \N and m \geq k then: S_k(m) = (-1)^k B_k^{(m+1)}(0), and: S_k(m)= {(-1)^k \over {m \choose k}} s_{m+1, m+1-k}.

• 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} \[6pt] &= \sum_{n=0}^k (-1)^n s_{k+n+1,n+1} \[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{align} Here, L_n^{(\alpha)} are Laguerre polynomials.

• The 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), {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 S_k^\prime(x) = -\sum_{j=0}^{k-1} {k \choose j} S_j(x) \frac{B_{k-j}}{k-j}. s_{n,m}= (-1)^{n-m} {n-1 \choose n-m} S_{n-m}(n-1). Conversely, S_{n,m}=(-1)^{n-m} {n \choose m} S_{n-m}(-m-1); --

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

The article contains definitions and properties of special convolution polynomial families defined by special generating functions of the form F(z)^x for F(0)=1. Special cases of these convolution polynomial sequences include the binomial power series, \mathcal{B}_t(z) = 1 + z \mathcal{B}_t(z)^t, so-termed tree polynomials, the Bell numbers, B(n), and the Laguerre polynomials. For F_n(x) := [z^n] F(z)^x, the polynomials n! \cdot F_n(x) are said to be of binomial type, and moreover, satisfy the generating function relation \frac{z F_n(x+tn)}{(x+tn)} = [z^n] \mathcal{F}_t(z)^x for all t \in \mathbb{C}, where \mathcal{F}_t(z) is implicitly defined by a functional equation of the form \mathcal{F}_t(z) = F\left(x \mathcal{F}_t(z)^t\right). The article also discusses asymptotic approximations and methods applied to polynomial sequences of this type. and in the Concrete Mathematics reference. We first define these polynomials through the Stirling numbers of the first kind as

:\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

:(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, \scriptstyle{\left[\begin{matrix} x \ x-n \end{matrix} \right]} and \scriptstyle{\left{\begin{matrix} x \ x-n \end{matrix} \right}}, for integers n \geq 0 and arbitrary complex values of x. The next table provides several special cases of these Stirling polynomials for the first few n \geq 0.

:{| class="wikitable" style="text-align: left;" ! n !! σn(x) |- | 0 || \frac{1}{x} |- | 1 || \frac{1}{2} |- | 2 || \frac1{24} (3 x - 1) |- | 3 || \frac1{48} (x^2 - x) |- | 4 || \frac1{5760} (15 x^3 - 30 x^2 + 5 x + 2) |- | 5 || \frac1{11520} (3 x^4 - 10 x^3 + 5 x^2 + 2 x) |- | 6 || \frac1{2903040} (63 x^5 - 315 x^4 + 315 x^3 + 91 x^2 - 42 x - 16) |- | 7 || \frac1{5806080} (9 x^6 - 63 x^5 + 105 x^4 + 7 x^3 - 42 x^2 - 16 x) |- | 8 || \frac1{1393459200} (135 x^7 - 1260 x^6 + 3150 x^5 - 840 x^4 - 2345 x^3 - 540 x^2 + 404 x + 144) |- | 9 || \frac1{2786918400} (15 x^8 - 180 x^7 + 630 x^6 - 448 x^5 - 665 x^4 + 100 x^3 + 404 x^2 + 144 x) |- | 10 || \frac1{367873228800} (99 x^9 - 1485 x^8 + 6930 x^7 - 8778 x^6 - 8085 x^5 + 8195 x^4 + 11792 x^3 + 2068 x^2 - 2288 x - 768) |- |}

Generating functions

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

: \begin{align} \left(\frac{z e^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{align}

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

:\mathcal{S}t(z)^x = \sum{n \geq 0} x \sigma_n(x+tn) z^n.

We also have the related series identity

:\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

:\sum_{n \geq 0} (-1)^{n+1} m \cdot \sigma_n(n-m) z^n = \left(\frac{z}{\ln(1+z)}\right)^m

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

:(r+s) \sigma_n(r+s+tn) = rs \sum_{k=0}^n \sigma_k(r+tk) \sigma_{n-k}(s+t(n-k))

and

:n \sigma_n(r+s+tn) = s \sum_{k=0}^n k \sigma_k(r+tk) \sigma_{n-k}(s+t(n-k)).

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

: \begin{align} \left{\begin{matrix} n \ m \end{matrix} \right} & = (-1)^{n-m+1} \frac{n!}{(m-1)!} \sigma_{n-m}(-m)\ (\text{when } m \left[\begin{matrix} n \ m \end{matrix} \right] & = \frac{n!}{(m-1)!} \sigma_{n-m}(n)\ (\text{when } m n), \end{align}

and their relations to the Bernoulli numbers given by

: \begin{align} \sigma_n(m) & = \frac{(-1)^{m+n-1}}{m! (n-m)!} \sum_{0 \leq k 0 \ \sigma_n(m) & = -\frac{B_n}{n \cdot n!},\ m = 0. \end{align}

References

References

1. See section 4.8.8 of ''The Umbral Calculus'' (1984) reference linked below.
2. See [http://mathworld.wolfram.com/NorlundPolynomial.html Norlund polynomials] on MathWorld.
3. Gessel. (1978). "Stirling polynomials". J. Combin. Theory Ser. A.
4. Section 4.4.8 of ''The Umbral Calculus''.
5. Section 7.4 of ''Concrete Mathematics''.