Formula:KLS:09.05:07

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x ⁢ Q ^ n ⁡ ( x ) = Q ^ n + 1 ⁡ ( x ) + ( A n + C n ) ⁢ Q ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ Q ^ n - 1 ⁡ ( x ) 𝑥 Hahn-polynomial-monic-p 𝑛 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-monic-p 𝑛 1 𝑥 𝛼 𝛽 𝑁 subscript 𝐴 𝑛 subscript 𝐶 𝑛 Hahn-polynomial-monic-p 𝑛 𝑥 𝛼 𝛽 𝑁 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 Hahn-polynomial-monic-p 𝑛 1 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle x{\widehat{Q}}_{n}\!\left(x\right)=% {\widehat{Q}}_{n+1}\!\left(x\right)+\left(A_{n}+C_{n}\right){\widehat{Q}}_{n}% \!\left(x\right)+A_{n-1}C_{n}{\widehat{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{\widehat{Q}}_{n}}}}  : monic Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:monicHahn

Bibliography

Equation in Section 9.5 of KLS.

URL links

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