Formula:KLS:14.19:10

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x ⁢ P ^ n ( α ) ⁡ ( x ) = P ^ n + 1 ( α ) ⁡ ( x ) + 1 2 ⁢ q n + 1 2 ⁢ α + 1 4 ⁢ ( 1 + q 1 2 ) ⁢ P ^ n ( α ) ⁡ ( x ) + 1 4 ⁢ ( 1 - q n ) ⁢ ( 1 - q n + α ) ⁢ P ^ n - 1 ( α ) ⁡ ( x ) 𝑥 continuous-q-Laguerre-polynomial-monic-p 𝛼 𝑛 𝑥 𝑞 continuous-q-Laguerre-polynomial-monic-p 𝛼 𝑛 1 𝑥 𝑞 1 2 superscript 𝑞 𝑛 1 2 𝛼 1 4 1 superscript 𝑞 1 2 continuous-q-Laguerre-polynomial-monic-p 𝛼 𝑛 𝑥 𝑞 1 4 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝛼 continuous-q-Laguerre-polynomial-monic-p 𝛼 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}^{(\alpha)}_{n}\!% \left(x\right)={\widehat{P}}^{(\alpha)}_{n+1}\!\left(x\right)+\frac{1}{2}q^{n+% \frac{1}{2}\alpha+\frac{1}{4}}(1+q^{\frac{1}{2}}){\widehat{P}}^{(\alpha)}_{n}% \!\left(x\right){}+\frac{1}{4}(1-q^{n})(1-q^{n+\alpha}){\widehat{P}}^{(\alpha)% }_{n-1}\!\left(x\right)}}}

Proof

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Symbols List

P ^ n ( α ) subscript superscript ^ 𝑃 𝛼 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{P}}^{(\alpha)}_{n}}}}  : monic continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:monicctsqLaguerre

Bibliography

Equation in Section 14.19 of KLS.

URL links

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