哈恩多项式

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：

$Q_n(x;\alpha ,\beta ,N)={}_3{F}_2(-n,-x,n+\alpha +\beta +1;\alpha +1,-N+1;1).$

前几个哈恩多项式为

$h[5]:=1+27\ast x/(-4\ast \alpha -4)+3\ast x\ast \alpha /(-4\ast \alpha -4)+270\ast x^2/((-4\ast \alpha -4)\ast (-3\ast \alpha -6))+57\ast x^2\ast \alpha /((-4\ast \alpha -4)\ast (-3\ast \alpha -6))-270\ast x/((-4\ast \alpha -4)\ast (-3\ast \alpha -6))-57\ast x\ast \alpha /((-4\ast \alpha -4)\ast (-3\ast \alpha -6))+3\ast x^2\ast {\alpha }^2/((-4\ast \alpha -4)\ast (-3\ast \alpha -6))-3\ast x\ast {\alpha }^2/((-4\ast \alpha -4)\ast (-3\ast \alpha -6))+990\ast x^3/((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))+299\ast x^3\ast \alpha /((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))-2970\ast x^2/((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))-897\ast x^2\ast \alpha /((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))+30\ast x^3\ast {\alpha }^2/((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))-90\ast x^2\ast {\alpha }^2/((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))+1980\ast x/((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))+598\ast x\ast \alpha /((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))+60\ast x\ast {\alpha }^2/((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))+x^3\ast {\alpha }^3/((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))-3\ast x^2\ast {\alpha }^3/((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))+2\ast x\ast {\alpha }^3/((-4\ast \alpha -4)\ast (-3\ast \alpha -6)\ast (-2\ast \alpha -6))$$h[6]:=1+27\ast x/(-5\ast \alpha -5)+3\ast x\ast \alpha /(-5\ast \alpha -5)+270\ast x^2/((-5\ast \alpha -5)\ast (-4\ast \alpha -8))+57\ast x^2\ast \alpha /((-5\ast \alpha -5)\ast (-4\ast \alpha -8))-270\ast x/((-5\ast \alpha -5)\ast (-4\ast \alpha -8))-57\ast x\ast \alpha /((-5\ast \alpha -5)\ast (-4\ast \alpha -8))+3\ast x^2\ast {\alpha }^2/((-5\ast \alpha -5)\ast (-4\ast \alpha -8))-3\ast x\ast {\alpha }^2/((-5\ast \alpha -5)\ast (-4\ast \alpha -8))+990\ast x^3/((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9))+299\ast x^3\ast \alpha /((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9))-2970\ast x^2/((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9))-897\ast x^2\ast \alpha /((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9))+30\ast x^3\ast {\alpha }^2/((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9))-90\ast x^2\ast {\alpha }^2/((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9))+1980\ast x/((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9))+598\ast x\ast \alpha /((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9))+60\ast x\ast {\alpha }^2/((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9))+x^3\ast {\alpha }^3/((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9))-3\ast x^2\ast {\alpha }^3/((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9))+2\ast x\ast {\alpha }^3/((-5\ast \alpha -5)\ast (-4\ast \alpha -8)\ast (-3\ast \alpha -9)).$

正交性

对于$\alpha$ > -1 和 $\beta$ > -1 以及 $\alpha$ < -N $\beta$ < -N,下列正交关系成立

$\sum _{x=0}^N(\genfrac{}{}{0ex}{}{\alpha +x}{x})$$(\genfrac{}{}{0ex}{}{\beta +N-x}{N-x}))\ast Q_m(x;\alpha ,\beta ,N)Q_n(x;\alpha ,\beta ,N)=\frac{(1{)}^n(n+\alpha +\beta +1{)}_{N+1}(\beta +1{)}_n\ast n!}{2n+\alpha +\beta +1)(\alpha +1{)}_n\ast (-N{)}_n\ast N!}\ast {\delta }_{mn}$

归递关系

哈恩多项式满足下列归递关系 $-x\ast Q_n(x)$=$A_n\ast Q_{n+1}(x)$-$(A_n+C_n)\ast Q_n(x)$*$C_n\ast Q_{n-1}(x)$

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

$\underset{\delta \to \mathrm{\infty }}{\lim }{R}_n(\lambda (x);\alpha ,\beta ,-N-1,\delta )=Q_n(x;\alpha ,\beta ,N),$

哈恩多项式→雅可比多项式

$\underset{N\to \mathrm{\infty }}{\lim }{Q}_n(Nx;\alpha ,\beta ,N)=\frac{P_n^{(\alpha ,\beta )}(1-2x)}{P_n^{(\alpha ,\beta )}(1)}$

参考文献

• Roelof Koekoek, Peter A.Lesky,ReneF.Swarttouw,Hypergeometric Orthogonal Polynomials ad Their q=Aalogues, Springer,2008.