Formula:KLS:14.09:06

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2 ⁢ x ⁢ P n ⁡ ( x ; a | q ) = ( 1 - q n + 1 ) ⁢ P n + 1 ⁡ ( x ; a | q ) + 2 ⁢ a ⁢ q n ⁢ cos ⁡ ϕ ⁢ P n ⁡ ( x ; a | q ) + ( 1 - a 2 ⁢ q n - 1 ) ⁢ P n - 1 ⁡ ( x ; a | q ) 2 𝑥 q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 1 superscript 𝑞 𝑛 1 q-Meixner-Pollaczek-polynomial-P 𝑛 1 𝑥 𝑎 𝑞 2 𝑎 superscript 𝑞 𝑛 italic-ϕ q-Meixner-Pollaczek-polynomial-P 𝑛 𝑥 𝑎 𝑞 1 superscript 𝑎 2 superscript 𝑞 𝑛 1 q-Meixner-Pollaczek-polynomial-P 𝑛 1 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle 2xP_{n}\!\left(x;a|q\right)=(1-q^{n% +1})P_{n+1}\!\left(x;a|q\right)+2aq^{n}\cos\phi P_{n}\!\left(x;a|q\right){}+(1% -a^{2}q^{n-1})P_{n-1}\!\left(x;a|q\right)}}}

Proof

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Symbols List

P n subscript 𝑃 𝑛 {\displaystyle{\displaystyle{\displaystyle P_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Meixner-Pollaczek polynomial : http://drmf.wmflabs.org/wiki/Definition:qMeixnerPollaczek
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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