Formula:KLS:09.05:03

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- x ⁢ Q n ⁡ ( x ) = A n ⁢ Q n + 1 ⁡ ( x ) - ( A n + C n ) ⁢ Q n ⁡ ( x ) + C n ⁢ Q n - 1 ⁡ ( x ) 𝑥 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 subscript 𝐴 𝑛 Hahn-polynomial-Q 𝑛 1 𝑥 𝛼 𝛽 𝑁 subscript 𝐴 𝑛 subscript 𝐶 𝑛 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 subscript 𝐶 𝑛 Hahn-polynomial-Q 𝑛 1 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle-xQ_{n}\!\left(x\right)=A_{n}Q_{n+1}% \!\left(x\right)-\left(A_{n}+C_{n}\right)Q_{n}\!\left(x\right)+C_{n}Q_{n-1}\!% \left(x\right)}}}

Substitution(s)

C n = n ⁢ ( n + α + β + N + 1 ) ⁢ ( n + β ) ( 2 ⁢ n + α + β ) ⁢ ( 2 ⁢ n + α + β + 1 ) subscript 𝐶 𝑛 𝑛 𝑛 𝛼 𝛽 𝑁 1 𝑛 𝛽 2 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{n(n+\alpha+\beta+N+1)(n% +\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}}}} &
A n = ( n + α + β + 1 ) ⁢ ( n + α + 1 ) ⁢ ( N - n ) ( 2 ⁢ n + α + β + 1 ) ⁢ ( 2 ⁢ n + α + β + 2 ) subscript 𝐴 𝑛 𝑛 𝛼 𝛽 1 𝑛 𝛼 1 𝑁 𝑛 2 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(n+\alpha+\beta+1)(n+% \alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}}}}


Proof

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Symbols List

& : logical and
Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle{\displaystyle Q_{n}}}}  : Hahn polynomial : http://dlmf.nist.gov/18.19#T1.t1.r3

Bibliography

Equation in Section 9.5 of KLS.

URL links

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