Formula:KLS:14.21:07

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

Proof

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Symbols List

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

Bibliography

Equation in Section 14.21 of KLS.

URL links

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