Formula:KLS:14.19:09

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

Proof

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Symbols List

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

Bibliography

Equation in Section 14.19 of KLS.

URL links

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