Formula:KLS:09.09:05: Difference between revisions

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Latest revision as of 08:36, 22 December 2019


x ⁢ P ^ n ⁡ ( x ) = P ^ n + 1 ⁡ ( x ) + ( N + 1 ) ⁢ ν ( n - N - 1 ) ⁢ ( n - N ) ⁢ P ^ n ⁡ ( x ) - n ⁢ ( n - 2 ⁢ N - 2 ) ⁢ ( n - N - 1 - i ⁢ ν ) ⁢ ( n - N - 1 + i ⁢ ν ) ( 2 ⁢ n - 2 ⁢ N - 3 ) ⁢ ( n - N - 1 ) 2 ⁢ ( 2 ⁢ n - 2 ⁢ N - 1 ) ⁢ P ^ n - 1 ⁡ ( x ) 𝑥 pseudo-Jacobi-polynomial-monic-p 𝑛 𝑥 𝜈 𝑁 pseudo-Jacobi-polynomial-monic-p 𝑛 1 𝑥 𝜈 𝑁 𝑁 1 𝜈 𝑛 𝑁 1 𝑛 𝑁 pseudo-Jacobi-polynomial-monic-p 𝑛 𝑥 𝜈 𝑁 𝑛 𝑛 2 𝑁 2 𝑛 𝑁 1 imaginary-unit 𝜈 𝑛 𝑁 1 imaginary-unit 𝜈 2 𝑛 2 𝑁 3 superscript 𝑛 𝑁 1 2 2 𝑛 2 𝑁 1 pseudo-Jacobi-polynomial-monic-p 𝑛 1 𝑥 𝜈 𝑁 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}_{n}\!\left(x\right)=% {\widehat{P}}_{n+1}\!\left(x\right)+\frac{(N+1)\nu}{(n-N-1)(n-N)}{\widehat{P}}% _{n}\!\left(x\right){}-\frac{n(n-2N-2)(n-N-1-\mathrm{i}\nu)(n-N-1+\mathrm{i}% \nu)}{(2n-2N-3)(n-N-1)^{2}(2n-2N-1)}{\widehat{P}}_{n-1}\!\left(x\right)}}}

Proof

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Symbols List

P ^ n subscript ^ 𝑃 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{P}}_{n}}}}  : monic pseudo-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:monicpseudoJacobi
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 9.9 of KLS.

URL links

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