查看“哈恩多项式”的源代码

[[File:Hahn polynomials.gif|thumb|300px|哈恩多项式]]
'''哈恩多项式'''是一个以德国数学家Wolfgang Hahn命名的正交多项式，由下列[[广义超几何函数]]定义：<ref>{{harvs|txt | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010|loc=14}} </ref>

 
:<math>Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).\ </math>

前几个哈恩多项式为
: <math>
 h[5] := 1+27*x/(-4*\alpha-4)+3*x*\alpha/(-4*\alpha-4)+270*x^2/((-4*\alpha-4)*(-3*\alpha-6))+57*x^2*\alpha/((-4*\alpha-4)*(-3*\alpha-6))-270*x/((-4*\alpha-4)*(-3*\alpha-6))-57*x*\alpha/((-4*\alpha-4)*(-3*\alpha-6))+3*x^2*\alpha^2/((-4*\alpha-4)*(-3*\alpha-6))-3*x*\alpha^2/((-4*\alpha-4)*(-3*\alpha-6))+990*x^3/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))+299*x^3*\alpha/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))-2970*x^2/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))-897*x^2*\alpha/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))+30*x^3*\alpha^2/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))-90*x^2*\alpha^2/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))+1980*x/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))+598*x*\alpha/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))+60*x*\alpha^2/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))+x^3*\alpha^3/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))-3*x^2*\alpha^3/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))+2*x*\alpha^3/((-4*\alpha-4)*(-3*\alpha-6)*(-2*\alpha-6))</math><math>
h[6] := 1+27*x/(-5*\alpha-5)+3*x*\alpha/(-5*\alpha-5)+270*x^2/((-5*\alpha-5)*(-4*\alpha-8))+57*x^2*\alpha/((-5*\alpha-5)*(-4*\alpha-8))-270*x/((-5*\alpha-5)*(-4*\alpha-8))-57*x*\alpha/((-5*\alpha-5)*(-4*\alpha-8))+3*x^2*\alpha^2/((-5*\alpha-5)*(-4*\alpha-8))-3*x*\alpha^2/((-5*\alpha-5)*(-4*\alpha-8))+990*x^3/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9))+299*x^3*\alpha/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9))-2970*x^2/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9))-897*x^2*\alpha/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9))+30*x^3*\alpha^2/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9))-90*x^2*\alpha^2/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9))+1980*x/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9))+598*x*\alpha/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9))+60*x*\alpha^2/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9))+x^3*\alpha^3/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9))-3*x^2*\alpha^3/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9))+2*x*\alpha^3/((-5*\alpha-5)*(-4*\alpha-8)*(-3*\alpha-9)) .
</math>
==正交性==
对于<math> \alpha </math> > -1   和 <math> \beta </math> > -1  以及 <math>  \alpha </math> < -N 
<math>  \beta </math>  <  -N,下列正交关系成立<ref name=RO>Roelof Koekoek, p204</ref>

<math>\sum_{x=0}^{N} {  \alpha+x  \choose  x}</math><math>{\beta+N-x \choose N-x})*Q_m(x;\alpha,\beta,N)Q_n(x;\alpha,\beta,N)=\frac{(1)^n(n+\alpha+\beta+1)_{N+1}(\beta+1)_n*n!}{2n+\alpha+\beta+1)(\alpha+1)_n*(-N)_n*N!}*\delta_{mn}</math>

==归递关系==
哈恩多项式满足下列归递关系<ref name=K>Roelof KoeKoek p204</ref>
<math>-x*Q_n(x)</math>=<math>A_{n}*Q_{n+1}(x)</math>-<math>(A_{n}+C_{n})*Q_{n}(x)</math>*<math>C_{n}*Q_{n-1}(x)</math>

其中<math>Q_{n}(x)=Q_{n}(x;\alpha,\beta,N)</math>

==极限关系==
;[[拉卡多项式]]→'''哈恩多项式'''<ref name=R>Roelof,p206-207</ref>

:<math> \lim_{\delta \to \infty}R_{n}(\lambda(x);\alpha,\beta,-N-1,\delta)=Q_{n}(x;\alpha,\beta,N), \, </math>
;'''哈恩多项式'''→[[雅可比多项式]]

<math> \lim_{N \to \infty}Q_{n}(Nx;\alpha,\beta,N)=\frac{P_{n}^{(\alpha,\beta)}(1-2x) }{P_{n}^{(\alpha,\beta)}(1)}                                                </math>

==参考文献==
<references/>
*Roelof  Koekoek, Peter A.Lesky,ReneF.Swarttouw,Hypergeometric Orthogonal Polynomials ad Their q=Aalogues, Springer,2008.
[[Category:特殊超几何函数]]
[[Category:正交多项式]]