哈恩多项式

哈恩多项式（Hahn polynomials）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：

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

前几个哈恩多项式为
:
h[5] := 1+27x/(-4\alpha-4)+3x\alpha/(-4\alpha-4)+270x^2/((-4\alpha-4)(-3\alpha-6))+57x^2\alpha/((-4\alpha-4)(-3\alpha-6))-270x/((-4\alpha-4)(-3\alpha-6))-57x\alpha/((-4\alpha-4)(-3\alpha-6))+3x^2\alpha^2/((-4\alpha-4)(-3\alpha-6))-3x\alpha^2/((-4\alpha-4)(-3\alpha-6))+990x^3/((-4\alpha-4)(-3\alpha-6)(-2\alpha-6))+299x^3\alpha/((-4\alpha-4)(-3\alpha-6)(-2\alpha-6))-2970x^2/((-4\alpha-4)(-3\alpha-6)(-2\alpha-6))-897x^2\alpha/((-4\alpha-4)(-3\alpha-6)(-2\alpha-6))+30x^3\alpha^2/((-4\alpha-4)(-3\alpha-6)(-2\alpha-6))-90x^2\alpha^2/((-4\alpha-4)(-3\alpha-6)(-2\alpha-6))+1980x/((-4\alpha-4)(-3\alpha-6)(-2\alpha-6))+598x\alpha/((-4\alpha-4)(-3\alpha-6)(-2\alpha-6))+60x\alpha^2/((-4\alpha-4)(-3\alpha-6)(-2\alpha-6))+x^3\alpha^3/((-4\alpha-4)(-3\alpha-6)(-2\alpha-6))-3x^2\alpha^3/((-4\alpha-4)(-3\alpha-6)(-2\alpha-6))+2x\alpha^3/((-4\alpha-4)(-3\alpha-6)(-2*\alpha-6))
h[6] := 1+27x/(-5\alpha-5)+3x\alpha/(-5\alpha-5)+270x^2/((-5\alpha-5)(-4\alpha-8))+57x^2\alpha/((-5\alpha-5)(-4\alpha-8))-270x/((-5\alpha-5)(-4\alpha-8))-57x\alpha/((-5\alpha-5)(-4\alpha-8))+3x^2\alpha^2/((-5\alpha-5)(-4\alpha-8))-3x\alpha^2/((-5\alpha-5)(-4\alpha-8))+990x^3/((-5\alpha-5)(-4\alpha-8)(-3\alpha-9))+299x^3\alpha/((-5\alpha-5)(-4\alpha-8)(-3\alpha-9))-2970x^2/((-5\alpha-5)(-4\alpha-8)(-3\alpha-9))-897x^2\alpha/((-5\alpha-5)(-4\alpha-8)(-3\alpha-9))+30x^3\alpha^2/((-5\alpha-5)(-4\alpha-8)(-3\alpha-9))-90x^2\alpha^2/((-5\alpha-5)(-4\alpha-8)(-3\alpha-9))+1980x/((-5\alpha-5)(-4\alpha-8)(-3\alpha-9))+598x\alpha/((-5\alpha-5)(-4\alpha-8)(-3\alpha-9))+60x\alpha^2/((-5\alpha-5)(-4\alpha-8)(-3\alpha-9))+x^3\alpha^3/((-5\alpha-5)(-4\alpha-8)(-3\alpha-9))-3x^2\alpha^3/((-5\alpha-5)(-4\alpha-8)(-3\alpha-9))+2x\alpha^3/((-5\alpha-5)(-4\alpha-8)(-3*\alpha-9)) .

正交性
对于 \alpha > -1 和 \beta > -1 以及 \alpha \beta \sum_{x=0}^{N} { \alpha+x \choose x}{\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)_nn!}{2n+\alpha+\beta+1)(\alpha+1)_n(-N)_nN!}*\delta_{mn}

归递关系
哈恩多项式满足下列归递关系
-xQ_n(x)=A_{n}Q_{n+1}(x)-(A_{n}+C_{n})Q_{n}(x)C_{n}*Q_{n-1}(x)

其中Q_{n}(x)=Q_{n}(x;\alpha,\beta,N)

极限关系
;拉卡多项式→哈恩多项式

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

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

参考文献
*Roelof Koekoek, Peter A.Lesky,ReneF.Swarttouw,Hypergeometric Orthogonal Polynomials ad Their q=Aalogues, Springer,2008.