Formula:KLS:09.09:17

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P n ⁡ ( x ; ν , N ) = ( - 2 ⁢ i ) n ⁢ n ! ( n - 2 ⁢ N - 1 ) n ⁢ P n ( - N - 1 + i ⁢ ν , - N - 1 - i ⁢ ν ) ⁡ ( i ⁢ x ) pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 superscript 2 imaginary-unit 𝑛 𝑛 Pochhammer-symbol 𝑛 2 𝑁 1 𝑛 Jacobi-polynomial-P 𝑁 1 imaginary-unit 𝜈 𝑁 1 imaginary-unit 𝜈 𝑛 imaginary-unit 𝑥 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;\nu,N\right)=\frac{(% -2\mathrm{i})^{n}n!}{{\left(n-2N-1\right)_{n}}}P^{(-N-1+\mathrm{i}\nu,-N-1-% \mathrm{i}\nu)}_{n}\left(\mathrm{i}x\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
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
P n ( α , β ) subscript superscript 𝑃 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}  : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3

Bibliography

Equation in Section 9.9 of KLS.

URL links

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