Formula:KLS:09.09:12

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P n ⁡ ( x ; ν , N ) = ( 1 + x 2 ) N + 1 ⁢ e - 2 ⁢ ν ⁢ arctan ⁡ x ( n - 2 ⁢ N - 1 ) n ⁢ ( d d ⁢ x ) n ⁢ [ ( 1 + x 2 ) n - N - 1 ⁢ e 2 ⁢ ν ⁢ arctan ⁡ x ] pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 superscript 1 superscript 𝑥 2 𝑁 1 2 𝜈 𝑥 Pochhammer-symbol 𝑛 2 𝑁 1 𝑛 superscript 𝑑 𝑑 𝑥 𝑛 delimited-[] superscript 1 superscript 𝑥 2 𝑛 𝑁 1 2 𝜈 𝑥 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;\nu,N\right)=\frac{(% 1+x^{2})^{N+1}{\mathrm{e}^{-2\nu\operatorname{arctan}x}}}{{\left(n-2N-1\right)% _{n}}}\left(\frac{d}{dx}\right)^{n}\left[(1+x^{2})^{n-N-1}{\mathrm{e}^{2\nu% \operatorname{arctan}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
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii

Bibliography

Equation in Section 9.9 of KLS.

URL links

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