双対ハーン多項式
以下の母関数を持つ:
( 1 − t ) N − x 2 F 1 ( − x , − x − δ γ + 1 ; t ) = ∑ n = 0 N ( − N ) n n ! R n ( λ ( x ) ; γ , δ , N ) t n {\displaystyle (1-t)^{N-x}{_{2}F_{1}}\left({\begin{matrix}-x,-x-\delta \\\gamma +1\end{matrix}};t\right)=\sum _{n=0}^{N}{\frac {(-N)_{n}}{n!}}R_{n}(\lambda (x);\gamma ,\delta ,N)t^{n}} ( 1 − t ) x 2 F 1 ( x − N , x + γ + 1 − δ − N ; t ) = ∑ n = 0 N ( γ + 1 ) n ( − N ) n ( − δ − N ) n n ! R n ( λ ( x ) ; γ , δ , N ) t n {\displaystyle (1-t)^{x}{_{2}F_{1}}\left({\begin{matrix}x-N,x+\gamma +1\\-\delta -N\end{matrix}};t\right)=\sum _{n=0}^{N}{\frac {(\gamma +1)_{n}(-N)_{n}}{(-\delta -N)_{n}n!}}R_{n}(\lambda (x);\gamma ,\delta ,N)t^{n}} [ e t 2 F 2 ( − x , x + γ + δ + 1 γ + 1 , − N ; − t ) ] N = ∑ n = 0 N 1 n !