贝特曼多项式

贝特曼多项式(Bateman polynomials)是一个正交多项式，定义如下^[1]

贝特曼多项式图

${\displaystyle F_{n}\left({\frac {d}{dx}}\right)\cosh ^{-1}(x)=\cosh ^{-1}(x)P_{n}(\tanh(x))={}_{3}F_{2}(-n,n+1,(x+1)/2;1,1;1)}$

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

${\displaystyle F_{0}(x)=1}$;

${\displaystyle F_{1}(x)=-x}$;

${\displaystyle F_{2}(x)={\frac {1}{4}}+{\frac {3}{4}}x^{2}}$;

${\displaystyle F_{3}(x)=-{\frac {7}{12}}x-{\frac {5}{12}}x^{3}}$;

${\displaystyle F_{4}(x)={\frac {9}{64}}+{\frac {65}{96}}x^{2}+{\frac {35}{192}}x^{4}}$;

${\displaystyle F_{5}(x)={\frac {407}{960}}x-{\frac {49}{96}}x^{3}-{\frac {21}{320}}x^{5}}$;