Formula:KLS:14.01:11: Difference between revisions

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Latest revision as of 08:36, 22 December 2019


2 ⁢ x ⁢ p ~ n ⁡ ( x ) = A n ⁢ p ~ n + 1 ⁡ ( x ) + [ a + a - 1 - ( A n + C n ) ] ⁢ p ~ n ⁡ ( x ) + C n ⁢ p ~ n - 1 ⁡ ( x ) 2 𝑥 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 subscript 𝐴 𝑛 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 delimited-[] 𝑎 superscript 𝑎 1 subscript 𝐴 𝑛 subscript 𝐶 𝑛 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 subscript 𝐶 𝑛 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 {\displaystyle{\displaystyle{\displaystyle 2x{\tilde{p}}_{n}\!\left(x\right)=A% _{n}{\tilde{p}}_{n+1}\!\left(x\right)+\left[a+a^{-1}-\left(A_{n}+C_{n}\right)% \right]{\tilde{p}}_{n}\!\left(x\right)+C_{n}{\tilde{p}}_{n-1}\!\left(x\right)}}}

Substitution(s)

C n = a ⁢ ( 1 - q n ) ⁢ ( 1 - b ⁢ c ⁢ q n - 1 ) ⁢ ( 1 - b ⁢ d ⁢ q n - 1 ) ⁢ ( 1 - c ⁢ d ⁢ q n - 1 ) ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n - 2 ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n - 1 ) subscript 𝐶 𝑛 𝑎 1 superscript 𝑞 𝑛 1 𝑏 𝑐 superscript 𝑞 𝑛 1 1 𝑏 𝑑 superscript 𝑞 𝑛 1 1 𝑐 𝑑 superscript 𝑞 𝑛 1 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 2 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{a(1-q^{n})(1-bcq^{n-1})% (1-bdq^{n-1})(1-cdq^{n-1})}{(1-abcdq^{2n-2})(1-abcdq^{2n-1})}}}} &
A n = ( 1 - a ⁢ b ⁢ q n ) ⁢ ( 1 - a ⁢ c ⁢ q n ) ⁢ ( 1 - a ⁢ d ⁢ q n ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ) a ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n - 1 ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q 2 ⁢ n ) subscript 𝐴 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 1 𝑎 𝑐 superscript 𝑞 𝑛 1 𝑎 𝑑 superscript 𝑞 𝑛 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑎 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 1 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-abq^{n})(1-acq^{n})(% 1-adq^{n})(1-abcdq^{n-1})}{a(1-abcdq^{2n-1})(1-abcdq^{2n})}}}}


Proof

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Symbols List

& : logical and
p ~ n subscript ~ 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}}}}  : normalized Askey-Wilson polynomial p ~ ~ 𝑝 {\displaystyle{\displaystyle{\displaystyle{\tilde{p}}}}}  : http://drmf.wmflabs.org/wiki/Definition:normAskeyWilsonptilde

Bibliography

Equation in Section 14.1 of KLS.

URL links

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