ハーン多項式
以下の漸化式が成り立つ。
− x Q n ( x ) = A n Q n + 1 ( x ) − ( A n + C n ) Q n ( x ) + C n Q n − 1 ( x ) . {\displaystyle -xQ_{n}(x)=A_{n}Q_{n+1}(x)-(A_{n}+C_{n})Q_{n}(x)+C_{n}Q_{n-1}(x).}
但し、 Q n ( x ; α , β , N ) {\displaystyle Q_{n}(x;\alpha ,\beta ,N)} を Q n ( x ) {\displaystyle Q_{n}(x)} と略記し、
A n = ( n + α + β + 1 ) ( n + α + 1 ) ( N − n ) ( 2 n + α + β + 1 ) ( 2 n + α + β + 2 ) , C n = n ( n + α + β + N + 1 ) ( n + β ) ( 2 n + α + β ) ( 2 n + α + β + 1 ) {\displaystyle {\begin{aligned}A_{n}&={\frac {(n+\alpha +\beta +1)(n+\alpha +1)(N-n)}{(2n+\alpha +\beta +1)(2n+\alpha +\beta +2)}},\\C_{n}&={\frac {n(n+\alpha +\beta +N+1)(n+\beta )}{(2n+\alpha +\beta )(2n+\alpha +\beta +1)}}\end{aligned}}}
とした。
以下の母関数を持つ:
1 F 1 ( − x α + 1 ; − t ) 1 F 1 ( x − N β + 1 ; t ) = ∑ n = 0 N ( − N ) n ( β + 1 ) n n ! Q n ( x ; α , β , N ) t n {\displaystyle {_{1}F_{1}}\left({\begin{matrix}-x\\\alpha +1\end{matrix}};-t\right){_{1}F_{1}}\left({\begin{matrix}x-N\\\beta +1\end{matrix}};t\right)=\sum _{n=0}^{N}{\frac {(-N)_{n}}{(\beta +1)_{n}n!}}Q_{n}(x;\alpha ,\beta ,N)t^{n}} 2 F 0 ( − x , − x + β + N + 1 − ; − t ) 2 F 0 ( x − N , x + α + 1 − ; t ) = ∑ n = 0 N ( − N ) n ( α + 1 ) n n ! Q n ( x ; α , β , N ) t n {\displaystyle {_{2}F_{0}}\left({\begin{matrix}-x,-x+\beta +N+1\\-\end{matrix}};-t\right){_{2}F_{0}}\left({\begin{matrix}x-N,x+\alpha +1\\-\end{matrix}};t\right)=\sum _{n=0}^{N}{\frac {(-N)_{n}(\alpha +1)_{n}}{n!
}}Q_{n}(x;\alpha ,\beta ,N)t^{n}} [ ( 1 − t ) − α − β − 1 3 F 2 ( 1 2 ( α + β + 1 ) , 1 2 ( α + β + 2 ) , − x α + 1 , − N ; − 4 t ( 1 − t ) 2 ) ] N = ∑ n = 0 N ( α + β + 1 ) n n ! Q n ( x ; α , β , N ) t n {\displaystyle \left[(1-t)^{-\alpha -\beta -1}{_{3}F_{2}}\left({\begin{matrix}{\frac {1}{2}}(\alpha +\beta +1),{\frac {1}{2}}(\alpha +\beta +2),-x\\\alpha +1,-N\end{matrix}};-{\frac {4t}{(1-t)^{2}}}\right)\right]_{N}=\sum _{n=0}^{N}{\frac {(\alpha +\beta +1)_{n}}{n!}}Q_{n}(x;\alpha ,\beta ,N)t^{n}}