オイラー多項式の定義
オイラー多項式の定義
オイラー多項式\(E_{n}\left(x\right)\)を次で定義する。
\[ E_{n}\left(x\right)=\sum_{k=0}^{n}C\left(n,k\right)\frac{E_{k}}{2^{k}}\left(x-\frac{1}{2}\right)^{n-k} \] オイラー多項式は次の式でも表される。
\[ E_{n}\left(x\right)=\sum_{k=0}^{n}\frac{1}{2^{k}}\sum_{j=0}^{k}\left(-1\right)^{k}C\left(k,j\right)\left(x+j\right)^{n} \]
オイラー多項式\(E_{n}\left(x\right)\)を次で定義する。
\[ E_{n}\left(x\right)=\sum_{k=0}^{n}C\left(n,k\right)\frac{E_{k}}{2^{k}}\left(x-\frac{1}{2}\right)^{n-k} \] オイラー多項式は次の式でも表される。
\[ E_{n}\left(x\right)=\sum_{k=0}^{n}\frac{1}{2^{k}}\sum_{j=0}^{k}\left(-1\right)^{k}C\left(k,j\right)\left(x+j\right)^{n} \]
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\(E_{k}\)はオイラー数オイラー多項式の一覧
\[ \begin{array}{|c|c|} \hline n & E_{n}\left(x\right)\\ \hline 0 & 1\\ \hline 1 & x-\frac{1}{2}\\ \hline 2 & x^{2}-x\\ \hline 3 & x^{3}-\frac{3x^{2}}{2}+\frac{1}{4}\\ \hline 4 & x^{4}-2x^{3}+x\\ \hline 5 & x^{5}-\frac{5x^{4}}{2}+\frac{5x^{2}}{2}-\frac{1}{2}\\ \hline 6 & x^{6}-3x^{5}+5x^{3}-3x\\ \hline 7 & x^{7}-\frac{7x^{6}}{2}+\frac{35x^{4}}{4}-\frac{21x^{2}}{2}+\frac{17}{8}\\ \hline 8 & x^{8}-4x^{7}+14x^{5}-28x^{3}+17x\\ \hline 9 & x^{9}-\frac{9x^{8}}{2}+21x^{6}-63x^{4}+\frac{153x^{2}}{2}-\frac{31}{2}\\ \hline 10 & x^{10}-5x^{9}+30x^{7}-126x^{5}+255x^{3}-155x\\ \hline 11 & x^{11}-\frac{11x^{10}}{2}+\frac{165x^{8}}{4}-231x^{6}+\frac{2805x^{4}}{4}-\frac{1705x^{2}}{2}+\frac{691}{4}\\ \hline 12 & x^{12}-6x^{11}+55x^{9}-396x^{7}+1683x^{5}-3410x^{3}+2073x\\ \hline 13 & x^{13}-\frac{13x^{12}}{2}+\frac{143x^{10}}{2}-\frac{1287x^{8}}{2}+\frac{7293x^{6}}{2}-\frac{22165x^{4}}{2}+\frac{26949x^{2}}{2}-\frac{5461}{2}\\ \hline 14 & x^{14}-7x^{13}+91x^{11}-1001x^{9}+7293x^{7}-31031x^{5}+62881x^{3}-38227x\\ \hline 15 & x^{15}-\frac{15x^{14}}{2}+\frac{455x^{12}}{4}-\frac{3003x^{10}}{2}+\frac{109395x^{8}}{8}-\frac{155155x^{6}}{2}+\frac{943215x^{4}}{4}-\frac{573405x^{2}}{2}+\frac{929569}{16}\\ \hline 16 & x^{16}-8x^{15}+140x^{13}-2184x^{11}+24310x^{9}-177320x^{7}+754572x^{5}-1529080x^{3}+929569x\\ \hline 17 & x^{17}-\frac{17x^{16}}{2}+170x^{14}-3094x^{12}+41327x^{10}-376805x^{8}+2137954x^{6}-6498590x^{4}+\frac{15802673x^{2}}{2}-\frac{3202291}{2}\\ \hline 18 & x^{18}-9x^{17}+204x^{15}-4284x^{13}+67626x^{11}-753610x^{9}+5497596x^{7}-23394924x^{5}+47408019x^{3}-28820619x\\ \hline 19 & x^{19}-\frac{19x^{18}}{2}+\frac{969x^{16}}{4}-5814x^{14}+\frac{214149x^{12}}{2}-1431859x^{10}+\frac{26113581x^{8}}{2}-74083926x^{6}+\frac{900752361x^{4}}{4}-\frac{547591761x^{2}}{2}+\frac{221930581}{4}\\ \hline 20 & x^{20}-10x^{19}+285x^{17}-7752x^{15}+164730x^{13}-2603380x^{11}+29015090x^{9}-211668360x^{7}+900752361x^{5}-1825305870x^{3}+1109652905x \\\hline \end{array} \]
\[ \begin{array}{|c|c|} \hline n & E_{n}\left(x\right)\\ \hline 0 & 1\\ \hline 1 & x-\frac{1}{2}\\ \hline 2 & x^{2}-x\\ \hline 3 & x^{3}-\frac{3x^{2}}{2}+\frac{1}{4}\\ \hline 4 & x^{4}-2x^{3}+x\\ \hline 5 & x^{5}-\frac{5x^{4}}{2}+\frac{5x^{2}}{2}-\frac{1}{2}\\ \hline 6 & x^{6}-3x^{5}+5x^{3}-3x\\ \hline 7 & x^{7}-\frac{7x^{6}}{2}+\frac{35x^{4}}{4}-\frac{21x^{2}}{2}+\frac{17}{8}\\ \hline 8 & x^{8}-4x^{7}+14x^{5}-28x^{3}+17x\\ \hline 9 & x^{9}-\frac{9x^{8}}{2}+21x^{6}-63x^{4}+\frac{153x^{2}}{2}-\frac{31}{2}\\ \hline 10 & x^{10}-5x^{9}+30x^{7}-126x^{5}+255x^{3}-155x\\ \hline 11 & x^{11}-\frac{11x^{10}}{2}+\frac{165x^{8}}{4}-231x^{6}+\frac{2805x^{4}}{4}-\frac{1705x^{2}}{2}+\frac{691}{4}\\ \hline 12 & x^{12}-6x^{11}+55x^{9}-396x^{7}+1683x^{5}-3410x^{3}+2073x\\ \hline 13 & x^{13}-\frac{13x^{12}}{2}+\frac{143x^{10}}{2}-\frac{1287x^{8}}{2}+\frac{7293x^{6}}{2}-\frac{22165x^{4}}{2}+\frac{26949x^{2}}{2}-\frac{5461}{2}\\ \hline 14 & x^{14}-7x^{13}+91x^{11}-1001x^{9}+7293x^{7}-31031x^{5}+62881x^{3}-38227x\\ \hline 15 & x^{15}-\frac{15x^{14}}{2}+\frac{455x^{12}}{4}-\frac{3003x^{10}}{2}+\frac{109395x^{8}}{8}-\frac{155155x^{6}}{2}+\frac{943215x^{4}}{4}-\frac{573405x^{2}}{2}+\frac{929569}{16}\\ \hline 16 & x^{16}-8x^{15}+140x^{13}-2184x^{11}+24310x^{9}-177320x^{7}+754572x^{5}-1529080x^{3}+929569x\\ \hline 17 & x^{17}-\frac{17x^{16}}{2}+170x^{14}-3094x^{12}+41327x^{10}-376805x^{8}+2137954x^{6}-6498590x^{4}+\frac{15802673x^{2}}{2}-\frac{3202291}{2}\\ \hline 18 & x^{18}-9x^{17}+204x^{15}-4284x^{13}+67626x^{11}-753610x^{9}+5497596x^{7}-23394924x^{5}+47408019x^{3}-28820619x\\ \hline 19 & x^{19}-\frac{19x^{18}}{2}+\frac{969x^{16}}{4}-5814x^{14}+\frac{214149x^{12}}{2}-1431859x^{10}+\frac{26113581x^{8}}{2}-74083926x^{6}+\frac{900752361x^{4}}{4}-\frac{547591761x^{2}}{2}+\frac{221930581}{4}\\ \hline 20 & x^{20}-10x^{19}+285x^{17}-7752x^{15}+164730x^{13}-2603380x^{11}+29015090x^{9}-211668360x^{7}+900752361x^{5}-1825305870x^{3}+1109652905x \\\hline \end{array} \]
\[
\Delta=e^{D}-1
\]
とおくと、
\begin{align*} \Delta^{n}x^{m} & =\sum_{k=0}^{n}C\left(n,k\right)\left(-1\right)^{n-k}\left(e^{D}\right)^{k}x^{m}\\ & =\sum_{k=0}^{n}C\left(n,k\right)\left(-1\right)^{n-k}e^{kD}x^{m}\\ & =\sum_{k=0}^{n}C\left(n,k\right)\left(-1\right)^{n-k}\sum_{j=0}^{\infty}\frac{k^{j}D^{j}}{j!}x^{m}\\ & =\sum_{k=0}^{n}C\left(n,k\right)\left(-1\right)^{n-k}\sum_{j=0}^{\infty}\frac{k^{j}}{j!}P\left(m,j\right)x^{m}\\ & =\sum_{k=0}^{n}C\left(n,k\right)\left(-1\right)^{n-k}\sum_{j=0}^{\infty}k^{j}C\left(m,j\right)x^{m-j}\\ & =\sum_{k=0}^{n}\left(-1\right)^{n-k}C\left(n,k\right)\left(x+k\right)^{m} \end{align*} なので、
\begin{align*} E_{n}\left(x\right) & =\frac{2}{e^{D}+1}x^{n}\\ & =\frac{2}{2+\Delta}x^{n}\\ & =\frac{1}{1+\frac{\Delta}{2}}x^{n}\\ & =\sum_{k=0}^{\infty}\left(-\frac{\Delta}{2}\right)^{k}x^{n}\\ & =\sum_{k=0}^{\infty}\left(-\frac{1}{2}\right)^{k}\Delta^{k}x^{n}\\ & =\sum_{k=0}^{n}\left(-\frac{1}{2}\right)^{k}\Delta^{k}x^{n}\\ & =\sum_{k=0}^{n}\left(-\frac{1}{2}\right)^{k}\sum_{j=0}^{k}\left(-1\right)^{k-j}C\left(k,j\right)\left(x+j\right)^{n}\\ & =\sum_{k=0}^{n}\frac{1}{2^{k}}\sum_{j=0}^{k}\left(-1\right)^{j}C\left(k,j\right)\left(x+j\right)^{n} \end{align*} となる。
\begin{align*} \Delta^{n}x^{m} & =\sum_{k=0}^{n}C\left(n,k\right)\left(-1\right)^{n-k}\left(e^{D}\right)^{k}x^{m}\\ & =\sum_{k=0}^{n}C\left(n,k\right)\left(-1\right)^{n-k}e^{kD}x^{m}\\ & =\sum_{k=0}^{n}C\left(n,k\right)\left(-1\right)^{n-k}\sum_{j=0}^{\infty}\frac{k^{j}D^{j}}{j!}x^{m}\\ & =\sum_{k=0}^{n}C\left(n,k\right)\left(-1\right)^{n-k}\sum_{j=0}^{\infty}\frac{k^{j}}{j!}P\left(m,j\right)x^{m}\\ & =\sum_{k=0}^{n}C\left(n,k\right)\left(-1\right)^{n-k}\sum_{j=0}^{\infty}k^{j}C\left(m,j\right)x^{m-j}\\ & =\sum_{k=0}^{n}\left(-1\right)^{n-k}C\left(n,k\right)\left(x+k\right)^{m} \end{align*} なので、
\begin{align*} E_{n}\left(x\right) & =\frac{2}{e^{D}+1}x^{n}\\ & =\frac{2}{2+\Delta}x^{n}\\ & =\frac{1}{1+\frac{\Delta}{2}}x^{n}\\ & =\sum_{k=0}^{\infty}\left(-\frac{\Delta}{2}\right)^{k}x^{n}\\ & =\sum_{k=0}^{\infty}\left(-\frac{1}{2}\right)^{k}\Delta^{k}x^{n}\\ & =\sum_{k=0}^{n}\left(-\frac{1}{2}\right)^{k}\Delta^{k}x^{n}\\ & =\sum_{k=0}^{n}\left(-\frac{1}{2}\right)^{k}\sum_{j=0}^{k}\left(-1\right)^{k-j}C\left(k,j\right)\left(x+j\right)^{n}\\ & =\sum_{k=0}^{n}\frac{1}{2^{k}}\sum_{j=0}^{k}\left(-1\right)^{j}C\left(k,j\right)\left(x+j\right)^{n} \end{align*} となる。
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(*)オイラー多項式の特殊値
\[
E_{n}\left(\frac{1}{2}\right)=\frac{E_{n}}{2^{n}}
\]
(*)オイラー多項式の総和
\[
E_{n}\left(x+y\right)=\sum_{k=0}^{n}C\left(n,k\right)E_{k}\left(x\right)y^{n-k}
\]
オイラー多項式の指数型母関数
\[
\sum_{k=0}^{\infty}\frac{E_{k}\left(x\right)}{k!}t^{k}=\frac{2e^{xt}}{e^{t}+1}
\]
オイラー多項式の性質
\[
E_{n}\left(1-x\right)=\left(-1\right)^{n}E_{n}\left(x\right)
\]