Formula:KLS:09.09:04: Difference between revisions

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

Proof

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Symbols List

P n subscript 𝑃 𝑛 {\displaystyle{\displaystyle{\displaystyle P_{n}}}}  : pseudo Jacobi polynomomal : http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi
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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