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Stirling polynomials
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<h2 id="definition-and-examples">Definition and examples</h2> <p>For nonnegative <a href="/facts/Integer/PUwwrawx">integers</a> <i>k</i>, the Stirling polynomials, <i>S</i><i>k</i>(<i>x</i>), are a <a href="/facts/Sheffer_sequence/CpjBBWcM">Sheffer sequence</a> for ( g ( t ) , f ¯ ( t ) ) := ( e − t , log ( t 1 − e − t ) ) {\displaystyle (g(t),{\bar {f}}(t)):=\left(e^{-t},\log \left({\frac {t}{1-e^{-t}}}\right)\right)} <a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> defined by the exponential generating function </p> ( t 1 − e − t ) x + 1 = ∑ k = 0 ∞ S k ( x ) t k k ! . {\displaystyle \left({t \over {1-e^{-t}}}\right)^{x+1}=\sum _{k=0}^{\infty }S_{k}(x){t^{k} \over k!}.} <p>The Stirling polynomials are a special case of the Nørlund polynomials (or generalized Bernoulli polynomials) <a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> each with exponential generating function </p> ( t e t − 1 ) a e z t = ∑ k = 0 ∞ B k ( a ) ( z ) t k k ! , {\displaystyle \left({t \over {e^{t}-1}}\right)^{a}e^{zt}=\sum _{k=0}^{\infty }B_{k}^{(a)}(z){t^{k} \over k!},} <p>given by the relation S k ( x ) = B k ( x + 1 ) ( x + 1 ) {\displaystyle S_{k}(x)=B_{k}^{(x+1)}(x+1)} . </p><p>The first 10 Stirling polynomials are given in the following table: </p> <table><tbody><tr><th><i>k</i></th><th><i>Sk</i>(<i>x</i>)</th></tr><tr><td>0</td><td> 1 {\displaystyle 1} </td></tr><tr><td>1</td><td> 1 2 ( x + 1 ) {\displaystyle {\scriptstyle {\frac {1}{2}}}(x+1)} </td></tr><tr><td>2</td><td> 1 12 ( 3 x 2 + 5 x + 2 ) {\displaystyle {\scriptstyle {\frac {1}{12}}}(3x^{2}+5x+2)} </td></tr><tr><td>3</td><td> 1 8 ( x 3 + 2 x 2 + x ) {\displaystyle {\scriptstyle {\frac {1}{8}}}(x^{3}+2x^{2}+x)} </td></tr><tr><td>4</td><td> 1 240 ( 15 x 4 + 30 x 3 + 5 x 2 − 18 x − 8 ) {\displaystyle {\scriptstyle {\frac {1}{240}}}(15x^{4}+30x^{3}+5x^{2}-18x-8)} </td></tr><tr><td>5</td><td> 1 96 ( 3 x 5 + 5 x 4 − 5 x 3 − 13 x 2 − 6 x ) {\displaystyle {\scriptstyle {\frac {1}{96}}}(3x^{5}+5x^{4}-5x^{3}-13x^{2}-6x)} </td></tr><tr><td>6</td><td> 1 4032 ( 63 x 6 + 63 x 5 − 315 x 4 − 539 x 3 − 84 x 2 + 236 x + 96 ) {\displaystyle {\scriptstyle {\frac {1}{4032}}}(63x^{6}+63x^{5}-315x^{4}-539x^{3}-84x^{2}+236x+96)} </td></tr><tr><td>7</td><td> 1 1152 ( 9 x 7 − 84 x 5 − 98 x 4 + 91 x 3 + 194 x 2 + 80 x ) {\displaystyle {\scriptstyle {\frac {1}{1152}}}(9x^{7}-84x^{5}-98x^{4}+91x^{3}+194x^{2}+80x)} </td></tr><tr><td>8</td><td> 1 34560 ( 135 x 8 − 180 x 7 − 1890 x 6 − 840 x 5 + 6055 x 4 + 8140 x 3 + 884 x 2 − 3088 x − 1152 ) {\displaystyle {\scriptstyle {\frac {1}{34560}}}(135x^{8}-180x^{7}-1890x^{6}-840x^{5}+6055x^{4}+8140x^{3}+884x^{2}-3088x-1152)} </td></tr><tr><td>9</td><td> 1 7680 ( 15 x 9 − 45 x 8 − 270 x 7 + 182 x 6 + 1687 x 5 + 1395 x 4 − 1576 x 3 − 2684 x 2 − 1008 x ) {\displaystyle {\scriptstyle {\frac {1}{7680}}}(15x^{9}-45x^{8}-270x^{7}+182x^{6}+1687x^{5}+1395x^{4}-1576x^{3}-2684x^{2}-1008x)} </td></tr></tbody></table> <p>Yet another variant of the Stirling polynomials is considered in <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> (see also the subsection on <a href="/facts/Stirling_polynomial/XWrKj3G5">Stirling convolution polynomials</a> 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 ) {\displaystyle f_{k}(n):=S(n+k,n)} and g k ( n ) := c ( n , n − k ) {\displaystyle g_{k}(n):=c(n,n-k)} where c ( n , k ) := ( − 1 ) n − k s ( n , k ) {\displaystyle c(n,k):=(-1)^{n-k}s(n,k)} are the <i>unsigned</i> <a href="/facts/Stirling_numbers_of_the_first_kind/Twdk4kA5">Stirling numbers of the first kind</a>, in terms of the two <a href="/facts/Stirling_number/MC05d5pF">Stirling number</a> triangles for non-negative integers n ≥ 1 , k ≥ 0 {\displaystyle n\geq 1,\ k\geq 0} . For fixed k ≥ 0 {\displaystyle k\geq 0} , both f k ( n ) {\displaystyle f_{k}(n)} and g k ( n ) {\displaystyle g_{k}(n)} are polynomials of the input n ∈ Z + {\displaystyle n\in \mathbb {Z} ^{+}} each of degree 2 k {\displaystyle 2k} and with leading coefficient given by the <a href="/facts/Double_factorial/VSU8AgzB">double factorial</a> term ( 1 ⋅ 3 ⋅ 5 ⋯ ( 2 k − 1 ) ) / ( 2 k ) ! {\displaystyle (1\cdot 3\cdot 5\cdots (2k-1))/(2k)!} . </p> <h2 id="properties">Properties</h2> <p>Below B k ( x ) {\displaystyle B_{k}(x)} denote the <a href="/facts/Bernoulli_polynomials/o3JBxUTN">Bernoulli polynomials</a> and B k = B k ( 0 ) {\displaystyle B_{k}=B_{k}(0)} the <a href="/facts/Bernoulli_numbers/RH0mblhs">Bernoulli numbers</a> under the convention B 1 = B 1 ( 0 ) = − 1 2 ; {\displaystyle B_{1}=B_{1}(0)=-{\tfrac {1}{2}};} s m , n {\displaystyle s_{m,n}} denotes a <a href="/facts/Stirling_number_of_the_first_kind/Twdk4kA5">Stirling number of the first kind</a>; and S m , n {\displaystyle S_{m,n}} denotes <a href="/facts/Stirling_numbers_of_the_second_kind/qg5UrHmk">Stirling numbers of the second kind</a>. </p> <ul><li>Special values: S k ( − m ) = ( − 1 ) k ( k + m − 1 k ) S k + m − 1 , m − 1 0 < m ∈ Z S k ( − 1 ) = δ k , 0 S k ( 0 ) = ( − 1 ) k B k S k ( 1 ) = ( − 1 ) k + 1 ( ( k − 1 ) B k + k B k − 1 ) S k ( 2 ) = ( − 1 ) k 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 ! {\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}}} </li> <li>If m ∈ N {\displaystyle m\in \mathbb {N} } and m < k {\displaystyle m<k} then: S k ( m ) = ( m + 1 ) ( k m + 1 ) ∑ j = 0 m ( − 1 ) m − j s m + 1 , m + 1 − j B k − j k − j . {\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}}.} </li> <li>If m ∈ N {\displaystyle m\in \mathbb {N} } and m ≥ k {\displaystyle m\geq k} then:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> S k ( m ) = ( − 1 ) k B k ( m + 1 ) ( 0 ) , {\displaystyle S_{k}(m)=(-1)^{k}B_{k}^{(m+1)}(0),} and: S k ( m ) = ( − 1 ) k ( m k ) s m + 1 , m + 1 − k . {\displaystyle S_{k}(m)={(-1)^{k} \over {m \choose k}}s_{m+1,m+1-k}.} </li></ul> <ul><li>The sequence S k ( x − 1 ) {\displaystyle S_{k}(x-1)} is of <a href="/facts/Binomial_type/labomtgk">binomial type</a>, since S k ( x + y − 1 ) = ∑ i = 0 k ( k i ) S i ( x − 1 ) S k − i ( y − 1 ) . {\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: S k ( x ) = ( x − k ) S k ( x − 1 ) x + k S k − 1 ( x + 1 ) . {\displaystyle S_{k}(x)=(x-k){S_{k}(x-1) \over x}+kS_{k-1}(x+1).} </li> <li>Explicit representations involving Stirling numbers can be deduced with <a href="/facts/Lagrange_interpolation/MnpMmhtC">Lagrange's interpolation formula</a>: S k ( x ) = ∑ n = 0 k ( − 1 ) k − n S k + n , n ( x + n n ) ( x + k + 1 k − n ) ( k + n n ) = ∑ n = 0 k ( − 1 ) n s k + n + 1 , n + 1 ( x − k n ) ( x − k − n − 1 k − n ) ( k + n k ) = k ! ∑ j = 0 k ( − 1 ) k − j ∑ m = j k ( x + m m ) ( m j ) L k + m ( − k − j ) ( − j ) {\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, L n ( α ) {\displaystyle L_{n}^{(\alpha )}} are <a href="/facts/Laguerre_polynomials/7RM3ypWl">Laguerre polynomials</a>.</li> <li>The following relations hold as well: ( k + m k ) S k ( x − m ) = ∑ i = 0 k ( − 1 ) k − i ( k + m i ) S k − i + m , m ⋅ S i ( x ) , {\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),} ( k − m k ) S k ( x + m ) = ∑ i = 0 k ( k − m i ) s m , m − k + i ⋅ 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).} </li> <li>By differentiating the generating function it readily follows that S k ′ ( x ) = − ∑ j = 0 k − 1 ( k j ) S j ( x ) B k − j k − j . {\displaystyle S_{k}^{\prime }(x)=-\sum _{j=0}^{k-1}{k \choose j}S_{j}(x){\frac {B_{k-j}}{k-j}}.} </li></ul> <h2 id="stirling-convolution-polynomials">Stirling convolution polynomials</h2> <h3>Definition and examples</h3> <p>Another variant of the Stirling polynomial sequence corresponds to a special case of the convolution polynomials studied by Knuth's article <a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> and in the <i>Concrete Mathematics</i> reference. We first define these polynomials through the <a href="/facts/Stirling_numbers_of_the_first_kind/Twdk4kA5">Stirling numbers of the first kind</a> as </p> σ n ( x ) = [ x x − n ] ⋅ 1 x ( x − 1 ) ⋯ ( x − n ) . {\displaystyle \sigma _{n}(x)=\left[{\begin{matrix}x\\x-n\end{matrix}}\right]\cdot {\frac {1}{x(x-1)\cdots (x-n)}}.} <p>It follows that these polynomials satisfy the next recurrence relation given by </p> ( x + 1 ) σ n ( x + 1 ) = ( x − n ) σ n ( x ) + x σ n − 1 ( x ) , n ≥ 1. {\displaystyle (x+1)\sigma _{n}(x+1)=(x-n)\sigma _{n}(x)+x\sigma _{n-1}(x),\ n\geq 1.} <p>These Stirling "<i>convolution</i>" polynomials may be used to define the Stirling numbers, [ x x − n ] {\displaystyle \scriptstyle {\left[{\begin{matrix}x\\x-n\end{matrix}}\right]}} and { x x − n } {\displaystyle \scriptstyle {\left\{{\begin{matrix}x\\x-n\end{matrix}}\right\}}} , for integers n ≥ 0 {\displaystyle n\geq 0} and <i>arbitrary</i> complex values of x {\displaystyle x} . The next table provides several special cases of these Stirling polynomials for the first few n ≥ 0 {\displaystyle n\geq 0} . </p> <table><tbody><tr><th><i>n</i></th><th><i>σn</i>(<i>x</i>)</th></tr><tr><td>0</td><td> 1 x {\displaystyle {\frac {1}{x}}} </td></tr><tr><td>1</td><td> 1 2 {\displaystyle {\frac {1}{2}}} </td></tr><tr><td>2</td><td> 1 24 ( 3 x − 1 ) {\displaystyle {\frac {1}{24}}(3x-1)} </td></tr><tr><td>3</td><td> 1 48 ( x 2 − x ) {\displaystyle {\frac {1}{48}}(x^{2}-x)} </td></tr><tr><td>4</td><td> 1 5760 ( 15 x 3 − 30 x 2 + 5 x + 2 ) {\displaystyle {\frac {1}{5760}}(15x^{3}-30x^{2}+5x+2)} </td></tr><tr><td>5</td><td> 1 11520 ( 3 x 4 − 10 x 3 + 5 x 2 + 2 x ) {\displaystyle {\frac {1}{11520}}(3x^{4}-10x^{3}+5x^{2}+2x)} </td></tr><tr><td>6</td><td> 1 2903040 ( 63 x 5 − 315 x 4 + 315 x 3 + 91 x 2 − 42 x − 16 ) {\displaystyle {\frac {1}{2903040}}(63x^{5}-315x^{4}+315x^{3}+91x^{2}-42x-16)} </td></tr><tr><td>7</td><td> 1 5806080 ( 9 x 6 − 63 x 5 + 105 x 4 + 7 x 3 − 42 x 2 − 16 x ) {\displaystyle {\frac {1}{5806080}}(9x^{6}-63x^{5}+105x^{4}+7x^{3}-42x^{2}-16x)} </td></tr><tr><td>8</td><td> 1 1393459200 ( 135 x 7 − 1260 x 6 + 3150 x 5 − 840 x 4 − 2345 x 3 − 540 x 2 + 404 x + 144 ) {\displaystyle {\frac {1}{1393459200}}(135x^{7}-1260x^{6}+3150x^{5}-840x^{4}-2345x^{3}-540x^{2}+404x+144)} </td></tr><tr><td>9</td><td> 1 2786918400 ( 15 x 8 − 180 x 7 + 630 x 6 − 448 x 5 − 665 x 4 + 100 x 3 + 404 x 2 + 144 x ) {\displaystyle {\frac {1}{2786918400}}(15x^{8}-180x^{7}+630x^{6}-448x^{5}-665x^{4}+100x^{3}+404x^{2}+144x)} </td></tr><tr><td>10</td><td> 1 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 ) {\displaystyle {\frac {1}{367873228800}}(99x^{9}-1485x^{8}+6930x^{7}-8778x^{6}-8085x^{5}+8195x^{4}+11792x^{3}+2068x^{2}-2288x-768)} </td></tr></tbody></table> <h3>Generating functions</h3> <p>This variant of the Stirling polynomial sequence has particularly nice ordinary <a href="/facts/Generating_functions/K7UVjnjL">generating functions</a> of the following forms: </p> ( z e z e z − 1 ) x = ∑ n ≥ 0 x σ n ( x ) z n ( 1 z ln 1 1 − z ) x = ∑ n ≥ 0 x σ n ( x + n ) z n . {\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}}} <p>More generally, if S t ( z ) {\displaystyle {\mathcal {S}}_{t}(z)} is a <a href="/facts/Power_series/vYh6sTps">power series</a> that satisfies ln ( 1 − z S t ( z ) t − 1 ) = − z S t ( z ) t {\displaystyle \ln \left(1-z{\mathcal {S}}_{t}(z)^{t-1}\right)=-z{\mathcal {S}}_{t}(z)^{t}} , we have that </p> S t ( z ) x = ∑ n ≥ 0 x σ n ( x + t n ) z n . {\displaystyle {\mathcal {S}}_{t}(z)^{x}=\sum _{n\geq 0}x\sigma _{n}(x+tn)z^{n}.} <p>We also have the related series identity <a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> ∑ n ≥ 0 ( − 1 ) n − 1 σ n ( n − 1 ) z n = z ln ( 1 + z ) = 1 + z 2 − z 2 12 + ⋯ , {\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 ,} <p>and the Stirling (Sheffer) polynomial related generating functions given by </p> ∑ n ≥ 0 ( − 1 ) n + 1 m ⋅ σ n ( n − m ) z n = ( z ln ( 1 + z ) ) m {\displaystyle \sum _{n\geq 0}(-1)^{n+1}m\cdot \sigma _{n}(n-m)z^{n}=\left({\frac {z}{\ln(1+z)}}\right)^{m}} ∑ n ≥ 0 ( − 1 ) n + 1 m ⋅ σ n ( m ) z n = ( z 1 − e − z ) m . {\displaystyle \sum _{n\geq 0}(-1)^{n+1}m\cdot \sigma _{n}(m)z^{n}=\left({\frac {z}{1-e^{-z}}}\right)^{m}.} <h3>Properties and relations</h3> <p>For integers 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} and r , s ∈ C {\displaystyle r,s\in \mathbb {C} } , these polynomials satisfy the two Stirling convolution formulas given by </p> ( r + s ) σ n ( r + s + t n ) = r s ∑ k = 0 n σ k ( r + t k ) σ n − k ( s + t ( n − k ) ) {\displaystyle (r+s)\sigma _{n}(r+s+tn)=rs\sum _{k=0}^{n}\sigma _{k}(r+tk)\sigma _{n-k}(s+t(n-k))} <p>and </p> n σ n ( r + s + t n ) = s ∑ k = 0 n k σ k ( r + t k ) σ n − k ( s + t ( n − k ) ) . {\displaystyle n\sigma _{n}(r+s+tn)=s\sum _{k=0}^{n}k\sigma _{k}(r+tk)\sigma _{n-k}(s+t(n-k)).} <p>When n , m ∈ N {\displaystyle n,m\in \mathbb {N} } , we also have that the polynomials, σ n ( m ) {\displaystyle \sigma _{n}(m)} , are defined through their relations to the <a href="/facts/Stirling_numbers/MC05d5pF">Stirling numbers</a> </p> { n m } = ( − 1 ) n − m + 1 n ! ( m − 1 ) ! σ n − m ( − m ) ( when m < 0 ) [ n m ] = n ! ( m − 1 ) ! σ n − m ( n ) ( when m > n ) , {\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}}} <p>and their relations to the <a href="/facts/Bernoulli_numbers/RH0mblhs">Bernoulli numbers</a> given by </p> σ n ( m ) = ( − 1 ) m + n − 1 m ! ( n − m ) ! ∑ 0 ≤ k < m [ m m − k ] B n − k n − k , n ≥ m > 0 σ n ( m ) = − B n n ⋅ n ! , m = 0. {\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}}} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Bernoulli_polynomials/o3JBxUTN">Bernoulli polynomials</a></li> <li><a href="/facts/Bernoulli_polynomials_of_the_second_kind/nu5NvaS1">Bernoulli polynomials of the second kind</a></li> <li><a href="/facts/Sheffer_sequence/CpjBBWcM">Sheffer</a> and <a href="/facts/Appell_sequence/7hEeFjzM">Appell sequences</a></li> <li><a href="/facts/Difference_polynomials/Dd09q2NC">Difference polynomials</a></li> <li><a href="/facts/Generating_function/K7UVjnjL">Special polynomial generating functions</a></li> <li><a href="/facts/Gregory_coefficients/r2UzbsUa">Gregory coefficients</a></li></ul> <ul><li>Erdeli, A.; Magnus, W.; Oberhettinger, F. & Tricomi, F. G. <i>Higher Transcendental Functions. Volume III</i>. New York.</li> <li>Graham; Knuth & Patashnik (1994). <i>Concrete Mathematics: A Foundation for Computer Science</i>.</li> <li>S. Roman (1984). <i>The Umbral Calculus</i>.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/StirlingPolynomial.html">"Stirling Polynomial"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/NorlundPolynomial.html">"Nørlund Polynomial"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><i>This article incorporates material from Stirling polynomial on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>See section 4.8.8 of The Umbral Calculus (1984) reference linked below. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>See Norlund polynomials on MathWorld. <a href="http://mathworld.wolfram.com/NorlundPolynomial.html" target="_blank">http://mathworld.wolfram.com/NorlundPolynomial.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Gessel & Stanley (1978). "Stirling polynomials". J. Combin. Theory Ser. A. 53: 24–33. doi:10.1016/0097-3165(78)90042-0. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Section 4.4.8 of The Umbral Calculus. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Knuth, D. E. (1992). "Convolution Polynomials". Mathematica J. 2: 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 {\displaystyle F(z)^{x}} for F ( 0 ) = 1 {\displaystyle F(0)=1} . Special cases of these convolution polynomial sequences include the binomial power series, B t ( z ) = 1 + z B t ( z ) t {\displaystyle {\mathcal {B}}_{t}(z)=1+z{\mathcal {B}}_{t}(z)^{t}} , so-termed tree polynomials, the Bell numbers, B ( n ) {\displaystyle B(n)} , and the Laguerre polynomials. For F n ( x ) := [ z n ] F ( z ) x {\displaystyle F_{n}(x):=[z^{n}]F(z)^{x}} , the polynomials n ! ⋅ F n ( x ) {\displaystyle n!\cdot F_{n}(x)} are said to be of binomial type, and moreover, satisfy the generating function relation z F n ( x + t n ) ( x + t n ) = [ z n ] F t ( z ) x {\displaystyle {\frac {zF_{n}(x+tn)}{(x+tn)}}=[z^{n}]{\mathcal {F}}_{t}(z)^{x}} for all t ∈ C {\displaystyle t\in \mathbb {C} } , where F t ( z ) {\displaystyle {\mathcal {F}}_{t}(z)} is implicitly defined by a functional equation of the form F t ( z ) = F ( x F t ( z ) t ) {\displaystyle {\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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Section 7.4 of Concrete Mathematics. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>