Formula:KLS:14.11:07: Difference between revisions

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Latest revision as of 08:37, 22 December 2019


x ⁢ P ^ n ⁡ ( x ) = P ^ n + 1 ⁡ ( x ) + [ 1 - ( A n + C n ) ] ⁢ P ^ n ⁡ ( x ) - a ⁢ b ⁢ q n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ q n ) ⁢ ( 1 - b ⁢ q n ) ⁢ P ^ n - 1 ⁡ ( x ) 𝑥 big-q-Laguerre-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑞 big-q-Laguerre-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑞 delimited-[] 1 subscript 𝐴 𝑛 subscript 𝐶 𝑛 big-q-Laguerre-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑞 𝑎 𝑏 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 1 𝑏 superscript 𝑞 𝑛 big-q-Laguerre-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}_{n}\!\left(x\right)=% {\widehat{P}}_{n+1}\!\left(x\right)+\left[1-(A_{n}+C_{n})\right]{\widehat{P}}_% {n}\!\left(x\right){}-abq^{n+1}(1-q^{n})(1-aq^{n})(1-bq^{n}){\widehat{P}}_{n-1% }\!\left(x\right)}}}

Substitution(s)

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


Proof

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Symbols List

& : logical and
P ^ n subscript ^ 𝑃 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{P}}_{n}}}}  : monic big q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:monicbigqLaguerre

Bibliography

Equation in Section 14.11 of KLS.

URL links

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