查看“贝特曼多项式”的源代码

[[File:Bateman polynomials.gif|thumb|300px|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<ref>Bateman, H. (1933), "Some properties of a certain set of polynomials.", Tôhoku Mathematical Journal 37: 23–38</ref>

:<math>F_n\left(\frac{d}{dx}\right)\cosh^{-1}(x) = \cosh^{-1}(x)P_n(\tanh(x))
={}_3F_2(-n,n+1,(x+1)/2 ; 1,1; 1)</math>

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
:<math>F_0(x)=1</math>;
:<math>F_1(x)=-x</math>;
:<math>F_2(x)=\frac{1}{4}+\frac{3}{4}x^2</math>;
:<math>F_3(x)=-\frac{7}{12}x-\frac{5}{12}x^3</math>;
:<math>F_4(x)=\frac{9}{64}+\frac{65}{96}x^2+\frac{35}{192}x^4</math>;
:<math>F_5(x)=\frac{407}{960}x-\frac{49}{96}x^3-\frac{21}{320}x^5</math>;

==参考文献==
<references/>

*{{cite journal|first1= Nadhla A. |last1=Al-Salam
|title=A class of hypergeometric polynomials
|journal=Ann. Matem. Pura Applic.
|year=1967 | volume=75|number=1 | pages=95–120 | doi=10.1007/BF02416800
}}
*{{Citation | last1=Bateman | first1=H. | title=Some properties of a certain set of polynomials. | url=http://www.journalarchive.jst.go.jp/english/jnlabstract_en.php?cdjournal=tmj1911&cdvol=37&noissue=0&startpage=23 | jfm=59.0364.02 | year=1933 | journal=Tôhoku Mathematical Journal | volume=37 | pages=23–38 }}{{dead link|date=十月 2017 |bot=InternetArchiveBot }}
*{{Citation | last1=Carlitz | first1=Leonard | title=Some polynomials of Touchard connected with the Bernoulli numbers | url=http://cms.math.ca/10.4153/CJM-1957-021-9 | doi=10.4153/CJM-1957-021-9 | mr=0085361 | year=1957 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=9 | pages=188–190 | access-date=2015-01-14 | archive-url=https://web.archive.org/web/20120330060401/http://cms.math.ca/10.4153/CJM-1957-021-9 | archive-date=2012-03-30 | dead-url=yes }}
*{{Citation | last1=Koelink | first1=H. T. | title=On Jacobi and continuous Hahn polynomials | doi=10.1090/S0002-9939-96-03190-5 | mr=1307541 | year=1996 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=124 | issue=3 | pages=887–898}}
*{{Citation | last1=Pasternack | first1=Simon | title=A generalization of the polynomial F<sub>n</sub>(x) | mr=0000698 | year=1939 | journal=London, Edinburgh, Dublin Philosophical Magazine and Journal of Science | volume=28 | pages=209–226}}
*{{Citation | last1=Touchard | first1=Jacques | title=Nombres exponentiels et nombres de Bernoulli | url=http://cms.math.ca/10.4153/CJM-1956-034-1 | mr=0079021 | year=1956 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=8 | pages=305–320 | doi=10.4153/cjm-1956-034-1 | access-date=2015-01-14 | archive-url=https://web.archive.org/web/20120330060342/http://cms.math.ca/10.4153/CJM-1956-034-1 | archive-date=2012-03-30 | dead-url=yes }}



[[Category:正交多项式]]