贝特曼多项式维基百科,自由的 encyclopedia 贝特曼多项式(Bateman polynomials)是一个正交多项式,定义如下[1] F n ( d d x ) cosh − 1 ( x ) = cosh − 1 ( x ) P n ( tanh ( x ) ) = 3 F 2 ( − n , n + 1 , ( x + 1 ) / 2 ; 1 , 1 ; 1 ) {\displaystyle F_{n}\left({\frac {d}{dx}}\right)\cosh ^{-1}(x)=\cosh ^{-1}(x)P_{n}(\tanh(x))={}_{3}F_{2}(-n,n+1,(x+1)/2;1,1;1)} 贝特曼多项式图 其中 F为超几何函数,P是勒让得多项式 前几个贝特曼多项式为 F 0 ( x ) = 1 {\displaystyle F_{0}(x)=1} ; F 1 ( x ) = − x {\displaystyle F_{1}(x)=-x} ; F 2 ( x ) = 1 4 + 3 4 x 2 {\displaystyle F_{2}(x)={\frac {1}{4}}+{\frac {3}{4}}x^{2}} ; F 3 ( x ) = − 7 12 x − 5 12 x 3 {\displaystyle F_{3}(x)=-{\frac {7}{12}}x-{\frac {5}{12}}x^{3}} ; F 4 ( x ) = 9 64 + 65 96 x 2 + 35 192 x 4 {\displaystyle F_{4}(x)={\frac {9}{64}}+{\frac {65}{96}}x^{2}+{\frac {35}{192}}x^{4}} ; F 5 ( x ) = 407 960 x − 49 96 x 3 − 21 320 x 5 {\displaystyle F_{5}(x)={\frac {407}{960}}x-{\frac {49}{96}}x^{3}-{\frac {21}{320}}x^{5}} ;
贝特曼多项式(Bateman polynomials)是一个正交多项式,定义如下[1] F n ( d d x ) cosh − 1 ( x ) = cosh − 1 ( x ) P n ( tanh ( x ) ) = 3 F 2 ( − n , n + 1 , ( x + 1 ) / 2 ; 1 , 1 ; 1 ) {\displaystyle F_{n}\left({\frac {d}{dx}}\right)\cosh ^{-1}(x)=\cosh ^{-1}(x)P_{n}(\tanh(x))={}_{3}F_{2}(-n,n+1,(x+1)/2;1,1;1)} 贝特曼多项式图 其中 F为超几何函数,P是勒让得多项式 前几个贝特曼多项式为 F 0 ( x ) = 1 {\displaystyle F_{0}(x)=1} ; F 1 ( x ) = − x {\displaystyle F_{1}(x)=-x} ; F 2 ( x ) = 1 4 + 3 4 x 2 {\displaystyle F_{2}(x)={\frac {1}{4}}+{\frac {3}{4}}x^{2}} ; F 3 ( x ) = − 7 12 x − 5 12 x 3 {\displaystyle F_{3}(x)=-{\frac {7}{12}}x-{\frac {5}{12}}x^{3}} ; F 4 ( x ) = 9 64 + 65 96 x 2 + 35 192 x 4 {\displaystyle F_{4}(x)={\frac {9}{64}}+{\frac {65}{96}}x^{2}+{\frac {35}{192}}x^{4}} ; F 5 ( x ) = 407 960 x − 49 96 x 3 − 21 320 x 5 {\displaystyle F_{5}(x)={\frac {407}{960}}x-{\frac {49}{96}}x^{3}-{\frac {21}{320}}x^{5}} ;