Formula:KLS:14.20:07

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x ⁢ p ^ n ⁡ ( x ) = p ^ n + 1 ⁡ ( x ) + ( A n + C n ) ⁢ p ^ n ⁡ ( x ) + a ⁢ q 2 ⁢ n - 1 ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ q n ) ⁢ p ^ n - 1 ⁡ ( x ) 𝑥 little-q-Laguerre-Wall-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 little-q-Laguerre-Wall-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 little-q-Laguerre-Wall-polynomial-monic-p 𝑛 𝑥 𝑎 𝑞 𝑎 superscript 𝑞 2 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 little-q-Laguerre-Wall-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)+(A_{n}+C_{n}){\widehat{p}}_{n}\!\left(x% \right)+aq^{2n-1}(1-q^{n})(1-aq^{n}){\widehat{p}}_{n-1}\!\left(x\right)}}}

Substitution(s)

C n = a ⁢ q n ⁢ ( 1 - q n ) subscript 𝐶 𝑛 𝑎 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}=aq^{n}(1-q^{n})}}} &
A n = q n ⁢ ( 1 - a ⁢ q n + 1 ) subscript 𝐴 𝑛 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=q^{n}(1-aq^{n+1})}}}


Proof

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Symbols List

& : logical and
p ^ n subscript ^ 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{p}}_{n}}}}  : monic little q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Laguerre/Wall polynomial : http://drmf.wmflabs.org/wiki/Definition:moniclittleqLaguerre

Bibliography

Equation in Section 14.20 of KLS.

URL links

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