双哈恩多项式

Dual Hahn polynomials plot

Dual Hahn polynomials plot

双重哈恩多项式(Dual Hahn polynomials)是一个正交多项式，定义如下

$R_n(\lambda (x);\gamma ,\delta ,N)={}_3{F}_2(-n,-x,x+\gamma +\delta +1;\gamma +1,-N;1).$ 其中0≤n≤N

双重哈恩多项式的前几个：

正交性

双重哈恩多项式满足下列正交关系：

$\sum _{x=0}^N\frac{(2x+\gamma +\delta +1)(\gamma +1{)}_x(-N{)}_{x}N!}{(-1{)}^x(x+\gamma +\delta +1{)}_{N+1}(\delta +1{)}_{x}x!}$* $R_m(\lambda (x);\gamma ,\delta ,N)R_n(\lambda (x);\gamma ,\delta ,N)=\frac{{\delta }_{mn}}{(\genfrac{}{}{0ex}{}{\gamma +n}{n})(\genfrac{}{}{0ex}{}{\delta +N-n}{N-n})}$

极限关系

拉卡多项式→双重哈恩多项式

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

双重哈恩多项式→梅西纳多项式

$\underset{N\to \mathrm{\infty }}{\lim }{R}_n(\lambda (x);\beta -1,N(1-c)c^{-1},N)=M_n(x;\beta ,c)$