贝特曼多项式

贝特曼多项式图

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

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

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

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

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

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

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

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

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

参考文献

• Al-Salam, Nadhla A. A class of hypergeometric polynomials. Ann. Matem. Pura Applic. 1967, 75 (1): 95–120. doi:10.1007/BF02416800. 
• Bateman, H., Some properties of a certain set of polynomials., Tôhoku Mathematical Journal, 1933, 37: 23–38, JFM 59.0364.02 
• Carlitz, Leonard, Some polynomials of Touchard connected with the Bernoulli numbers, Canadian Journal of Mathematics, 1957, 9: 188–190, ISSN 0008-414X, MR 0085361, doi:10.4153/CJM-1957-021-9 
• Koelink, H. T., On Jacobi and continuous Hahn polynomials, Proceedings of the American Mathematical Society, 1996, 124 (3): 887–898, ISSN 0002-9939, MR 1307541, doi:10.1090/S0002-9939-96-03190-5 
• Pasternack, Simon, A generalization of the polynomial F_n(x), London, Edinburgh, Dublin Philosophical Magazine and Journal of Science, 1939, 28: 209–226, MR 0000698 
• Touchard, Jacques, Nombres exponentiels et nombres de Bernoulli, Canadian Journal of Mathematics, 1956, 8: 305–320, ISSN 0008-414X, MR 0079021, doi:10.4153/cjm-1956-034-1