雙哈恩多項式

雙重哈恩多項式(Dual Hahn polynomials)是一個正交多項式，定義如下^[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

雙重哈恩多項式的前幾個：

正交性

雙重哈恩多項式滿足下列正交關係：^[2]

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

極限關係

拉卡多項式→雙重哈恩多項式

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

雙重哈恩多項式→梅西納多項式

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