双対ハーン多項式
γ , δ < − 1 {\displaystyle \gamma ,\,\delta <-1} または γ , δ < − N {\displaystyle \gamma ,\,\delta <-N} に対して以下の直交関係を満たす:
∑ x = 0 N ( 2 x + γ + δ + 1 ) ( γ + 1 ) x ( − N ) x N ! ( − 1 ) x ( x + γ + δ + 1 ) N + 1 ( δ + 1 ) x N ! R m ( λ ( x ) ; γ , δ , N ) R n ( λ ( x ) ; γ , δ , N ) = δ m n ( γ + N n ) ( δ + N − n N − n ) . {\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}N!}}R_{m}(\lambda (x);\gamma ,\delta ,N)R_{n}(\lambda (x);\gamma ,\delta ,N)={\frac {\delta _{mn}}{{\binom {\gamma +N}{n}}{\binom {\delta +N-n}{N-n}}}}.}
但し、 ( a ) n {\displaystyle (a)_{n}} はポッホハマーの記号を表す。
ロドリゲスの公式に相当する以下の式を満たす:
ω ( x ; γ , δ , N ) R n ( λ ( x ) ) = ( γ + δ + 1 ) n ( ∇ ∇ λ ( x ) n ω ( x ; γ + n , δ , N − n ) ) . {\displaystyle \omega (x;\gamma ,\delta ,N)R_{n}(\lambda (x))=(\gamma +\delta +1)_{n}\left({\frac {\nabla }{\nabla \lambda (x)}}^{n}\omega (x;\gamma +n,\delta ,N-n)\right).} n
R n ( λ ( x ) ; γ , δ , N ) t n {\displaystyle \left[e^{t}{_{2}F_{2}}\left({\begin{matrix}-x,x+\gamma +\delta +1\\\gamma +1,-N\end{matrix}};-t\right)\right]_{N}=\sum _{n=0}^{N}{\frac {1}{n!}}R_{n}(\lambda (x);\gamma ,\delta ,N)t^{n}} [ ( 1 − t ) ε 3 F 2 ( ε , − x , x + γ + δ + 1 γ + 1 , − N ; t t − 1 ) ] N = ∑ n = 0 N ( ε ) n n ! R n ( λ ( x ) ; γ , δ , N ) t n ( ∀ ε ∈ R ) {\displaystyle \left[(1-t)^{\epsilon }{_{3}F_{2}}\left({\begin{matrix}\epsilon ,-x,x+\gamma +\delta +1\\\gamma +1,-N\end{matrix}};{\frac {t}{t-1}}\right)\right]_{N}=\sum _{n=0}^{N}{\frac {(\epsilon )_{n}}{n!}}R_{n}(\lambda (x);\gamma ,\delta ,N)t^{n}\quad (\forall \epsilon \in \mathbb {R} )}