Formula:KLS:14.09:07

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x ⁢ P ^ n ⁡ ( x ) = P ^ n + 1 ⁡ ( x ) + a ⁢ q n ⁢ cos ⁡ ϕ ⁢ P ^ n ⁡ ( x ) + 1 4 ⁢ ( 1 - q n ) ⁢ ( 1 - a 2 ⁢ q n - 1 ) ⁢ P ^ n - 1 ⁡ ( x ) 𝑥 q-Meixner-Pollaczek-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 q-Meixner-Pollaczek-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 𝑎 superscript 𝑞 𝑛 italic-ϕ q-Meixner-Pollaczek-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 1 4 1 superscript 𝑞 𝑛 1 superscript 𝑎 2 superscript 𝑞 𝑛 1 q-Meixner-Pollaczek-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}_{n}\!\left(x\right)=% {\widehat{P}}_{n+1}\!\left(x\right)+aq^{n}\cos\phi\,{\widehat{P}}_{n}\!\left(x% \right)+\frac{1}{4}(1-q^{n})(1-a^{2}q^{n-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 q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Meixner-Pollaczek polynomial : http://drmf.wmflabs.org/wiki/Definition:monicqMeixnerPollaczek
cos cos {\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}  : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.9 of KLS.

URL links

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