双哈恩多项式

双重哈恩多项式(Dual Hahn polynomials)是一个正交多项式，定义如下^[1]

${\displaystyle R_{n}(\lambda (x);\gamma ,\delta ,N)={}_{3}F_{2}(-n,-x,x+\gamma +\delta +1;\gamma +1,-N;1).}$ 其中0≤n≤N

Dual Hahn polynomials plot

Dual Hahn polynomials plot

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：^[2]

${\displaystyle \sum _{x=0}^{N}{\frac {(2x+\gamma +\delta +1)(\gamma +1)_{x}(-N)_{x}N!}{(-1)^{x}(x+\gamma +\delta +1)_{N+1}(\delta +1)_{x}x!}}}$* ${\displaystyle R_{m}(\lambda (x);\gamma ,\delta ,N)R_{n}(\lambda (x);\gamma ,\delta ,N)={\frac {\delta _{mn}}{{\gamma +n \choose n}{\delta +N-n \choose N-n}}}}$

极限关系

拉卡多项式→双重哈恩多项式

${\displaystyle \lim _{\beta \to \infty }R_{n}(\lambda (x);-N-1,\beta ,\gamma ,\delta )=R_{n}(\lambda (\lim _{\beta \to \infty }R_{n}(\lambda (x);-x);\gamma ,\delta ,N)}$

双重哈恩多项式→梅西纳多项式

${\displaystyle \lim _{N\to \infty }R_{n}(\lambda (x);\beta -1,N(1-c)c^{-1},N)=M_{n}(x;\beta ,c)}$