Formula:KLS:09.08:05

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x ⁢ P ^ n ( α , β ) ⁡ ( x ) ⁢ x = P ^ n + 1 ( α , β ) ⁡ ( x ) ⁢ x + β 2 - α 2 ( 2 ⁢ n + α + β ) ⁢ ( 2 ⁢ n + α + β + 2 ) ⁢ P ^ n ( α , β ) ⁡ ( x ) ⁢ x + 4 ⁢ n ⁢ ( n + α ) ⁢ ( n + β ) ⁢ ( n + α + β ) ( 2 ⁢ n + α + β - 1 ) ⁢ ( 2 ⁢ n + α + β ) 2 ⁢ ( 2 ⁢ n + α + β + 1 ) ⁢ P ^ n - 1 ( α , β ) ⁡ ( x ) ⁢ x 𝑥 Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 𝑥 𝑥 Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 1 𝑥 𝑥 superscript 𝛽 2 superscript 𝛼 2 2 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 2 Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 𝑥 𝑥 4 𝑛 𝑛 𝛼 𝑛 𝛽 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 1 superscript 2 𝑛 𝛼 𝛽 2 2 𝑛 𝛼 𝛽 1 Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}^{(\alpha,\beta)}_{n}% \left(x\right){x}={\widehat{P}}^{(\alpha,\beta)}_{n+1}\left(x\right){x}+\frac{% \beta^{2}-\alpha^{2}}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}{\widehat{P}}^{(% \alpha,\beta)}_{n}\left(x\right){x}{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+% \beta)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^{2}(2n+\alpha+\beta+1)}{\widehat{% P}}^{(\alpha,\beta)}_{n-1}\left(x\right){x}}}}

Constraint(s)

absent {\displaystyle{\displaystyle{\displaystyle\;\;}}}


Proof

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Symbols List

P ^ n ( α , β ) subscript superscript ^ 𝑃 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{P}}^{(\alpha,\beta)}_{n}}}}  : monic Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:monicJacobi

Bibliography

Equation in Section 9.8 of KLS.

URL links

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