Stirling polynomials

For nonnegative integers k, the Stirling polynomials, S_k(x), are a Sheffer sequence for $(g(t),\; \backslash bar\{f\}(t))\; :=\; \backslash left(e^\{-t\},\; \backslash log\backslash left(\backslash frac\{t\}\{1-e^\{-t\}\}\backslash right)\backslash right)$ See section 4.8.8 of The Umbral Calculus (1984) reference linked below. defined by the exponential generating function

::$\backslash left(\; \{t\; \backslash over\; \{1-e^\{-t\}\}\}\; \backslash right)\; ^\{x+1\}=\; \backslash sum\_\{k=0\}^\backslash infty\; S\_k(x)\{t^k\; \backslash over\; k!\}.$

The Stirling polynomials are a special case of the Nørlund polynomials (or generalized Bernoulli polynomials) See [http://mathworld.wolfram.com/NorlundPolynomial.html Norlund polynomials] on MathWorld. each with exponential generating function

::$\backslash left(\; \{t\; \backslash over\; \{e^t-1\}\}\; \backslash right)\; ^a\; e^\{z\; t\}=\; \backslash sum\_\{k=0\}^\backslash infty\; B^\{(a)\}\_k(z)\{t^k\; \backslash over\; k!\},$

given by the relation $S\_k(x)=\; B\_k^\{(x+1)\}(x+1)$.

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

:

Yet another variant of the Stirling polynomials is considered in {{cite journal|author=Gessel|author2=Stanley|name-list-style=amp|title=Stirling polynomials|journal=J. Combin. Theory Ser. A|date=1978|volume=53|pages=24–33|doi=10.1016/0097-3165(78)90042-0|doi-access=}} (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 $n\; \backslash geq\; 1,\backslash \; k\; \backslash geq\; 0$. For fixed $k\; \backslash geq\; 0$, both $f\_k(n)$ and $g\_k(n)$ are polynomials of the input $n\; \backslash in\; \backslash mathbb\{Z\}^\{+\}$ each of degree $2k$ and with leading coefficient given by the double factorial term $(1\; \backslash cdot\; 3\; \backslash cdot\; 5\; \backslash cdots\; (2k-1))\; /\; (2k)!$.

Below $B\_k(x)$ denote the Bernoulli polynomials and $B\_k\; =\; B\_k(0)$ the Bernoulli numbers under the convention $B\_1\; =\; B\_1(0)\; =\; -\backslash tfrac\{1\}\{2\};$ $s\_\{m,n\}$ denotes a Stirling number of the first kind; and $S\_\{m,n\}$ denotes Stirling numbers of the second kind.

• Special values: $$\backslash begin\{align\}$$

S_k(-m) &= \frac{(-1)^k}{{k+m-1 \choose k}} S_{k+m-1,m-1} && 0 < m \in \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+ 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 $m\; \backslash in\; \backslash N$ and then: $$S\_k(m)\; =\; \{(m+1)\}\backslash binom\; k\{m+1\}\; \backslash sum\_\{j=0\}^m(-1)^\{m-j\}\; s\_\{m+1,\; m+1-j\}\; \backslash frac\{B\_\{k-j\}\}\{k-j\}.$$
• If $m\; \backslash in\; \backslash N$ and $m\; \backslash geq\; k$ then:Section 4.4.8 of The Umbral Calculus. $$S\_k(m)\; =\; (-1)^k\; B\_k^\{(m+1)\}(0),$$ and: $$S\_k(m)=\; \{(-1)^k\; \backslash over\; \{m\; \backslash choose\; k\}\}\; s\_\{m+1,\; m+1-k\}.$$

• The sequence $S\_k(x-1)$ is of binomial type, since $$S\_k(x+y-1)=\; \backslash sum\_\{i=0\}^k\; \{k\; \backslash choose\; i\}\; S\_i(x-1)\; S\_\{k-i\}(y-1).$$ Moreover, this basic recursion holds: $$S\_k(x)=\; (x-k)\; \{S\_k(x-1)\; \backslash over\; x\}\; +\; k\; S\_\{k-1\}(x+1).$$
• Explicit representations involving Stirling numbers can be deduced with Lagrange's interpolation formula: $$\backslash begin\{align\}$$

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}} \\[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{align} Here, $L\_n^\{(\backslash alpha)\}$ are Laguerre polynomials.

• The following relations hold as well: $$\{k+m\; \backslash choose\; k\}\; S\_k(x-m)=\; \backslash sum\_\{i=0\}^k\; (-1)^\{k-i\}\; \{k+m\; \backslash choose\; i\}\; S\_\{k-i+m,m\}\; \backslash cdot\; S\_i(x),$$ $$\{k-m\; \backslash choose\; k\}\; S\_k(x+m)=\; \backslash sum\_\{i=0\}^k\; \{k-m\; \backslash choose\; i\}\; s\_\{m,m-k+i\}\; \backslash cdot\; S\_i(x).$$
• By differentiating the generating function it readily follows that $$S\_k^\backslash prime(x)\; =\; -\backslash sum\_\{j=0\}^\{k-1\}\; \{k\; \backslash choose\; j\}\; S\_j(x)\; \backslash frac\{B\_\{k-j\}\}\{k-j\}.$$

Another variant of the Stirling polynomial sequence corresponds to a special case of the convolution polynomials studied by Knuth's article {{cite journal|last1=Knuth|first1=D. E.|title=Convolution Polynomials|journal=Mathematica J.|date=1992|volume=2|pages=67–78|arxiv=math/9207221|bibcode=1992math......7221K}}

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, $\backslash mathcal\{B\}\_t(z)\; =\; 1\; +\; z\; \backslash mathcal\{B\}\_t(z)^t$, so-termed tree polynomials, the Bell numbers, $B(n)$, and the Laguerre polynomials. For $F\_n(x)\; :=\; [z^n]\; F(z)^x$, the polynomials $n!\; \backslash cdot\; F\_n(x)$ are said to be of binomial type, and moreover, satisfy the generating function relation $\backslash frac\{z\; F\_n(x+tn)\}\{(x+tn)\}\; =\; [z^n]\; \backslash mathcal\{F\}\_t(z)^x$ for all $t\; \backslash in\; \backslash mathbb\{C\}$, where $\backslash mathcal\{F\}\_t(z)$ is implicitly defined by a functional equation of the form $\backslash mathcal\{F\}\_t(z)\; =\; F\backslash left(x\; \backslash mathcal\{F\}\_t(z)^t\backslash right)$. 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

:$\backslash sigma\_n(x)\; =\; \backslash left[\backslash begin\{matrix\}\; x\; \backslash \backslash \; x-n\; \backslash end\{matrix\}\; \backslash right]\; \backslash cdot\; \backslash frac\{1\}\{x(x-1)\backslash cdots(x-n)\}.$

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

:$(x+1)\; \backslash sigma\_n(x+1)\; =\; (x-n)\; \backslash sigma\_n(x)\; +\; x\; \backslash sigma\_\{n-1\}(x),\backslash \; n\; \backslash geq\; 1.$

These Stirling "convolution" polynomials may be used to define the Stirling numbers, $\backslash scriptstyle\{\backslash left[\backslash begin\{matrix\}\; x\; \backslash \backslash \; x-n\; \backslash end\{matrix\}\; \backslash right]\}$ and

$\backslash scriptstyle\{\backslash left\backslash \{\backslash begin\{matrix\}\; x\; \backslash \backslash \; x-n\; \backslash end\{matrix\}\; \backslash right\backslash \}\}$, for integers $n\; \backslash geq\; 0$ and arbitrary complex values of $x$.

The next table provides several special cases of these Stirling polynomials for the first few $n\; \backslash geq\; 0$.

:

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

:$\backslash mathcal\{S\}\_t(z)^x\; =\; \backslash sum\_\{n\; \backslash geq\; 0\}\; x\; \backslash sigma\_n(x+tn)\; z^n.$

We also have the related series identity Section 7.4 of Concrete Mathematics.

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

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

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

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

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

:$(r+s)\; \backslash sigma\_n(r+s+tn)\; =\; rs\; \backslash sum\_\{k=0\}^n\; \backslash sigma\_k(r+tk)\; \backslash sigma\_\{n-k\}(s+t(n-k))$

and

:$n\; \backslash sigma\_n(r+s+tn)\; =\; s\; \backslash sum\_\{k=0\}^n\; k\; \backslash sigma\_k(r+tk)\; \backslash sigma\_\{n-k\}(s+t(n-k)).$

When $n,\; m\; \backslash in\; \backslash mathbb\{N\}$, we also have that the polynomials, $\backslash sigma\_n(m)$, 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 < 0) \\

\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 < 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{align}

• Bernoulli polynomials
• Bernoulli polynomials of the second kind
• Sheffer and Appell sequences
• Difference polynomials
• Special polynomial generating functions
• Gregory coefficients

{{Reflist}}

• {{cite book | author=Erdeli, A. | author2=Magnus, W. | author3=Oberhettinger, F. | author4=Tricomi, F. G. | name-list-style=amp | title=Higher Transcendental Functions. Volume III | location=New York }}
• {{cite book | author1=Graham | author2=Knuth | author3=Patashnik | name-list-style=amp | title=Concrete Mathematics: A Foundation for Computer Science | year=1994 }}
• {{cite book | author=S. Roman | title=The Umbral Calculus | year=1984 }}

• {{MathWorld|urlname=StirlingPolynomial|title=Stirling Polynomial}}
• {{MathWorld|urlname=NorlundPolynomial|title=Nørlund Polynomial}}
• {{PlanetMath attribution|id=37575|title=Stirling polynomial}}

Category:Polynomials