Formula:KLS:14.05:06

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x ⁢ P ^ n ⁡ ( x ) = P ^ n + 1 ⁡ ( x ) + [ 1 - ( A n + C n ) ] ⁢ P ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ P ^ n - 1 ⁡ ( x ) 𝑥 big-q-Jacobi-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 big-q-Jacobi-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑞 delimited-[] 1 subscript 𝐴 𝑛 subscript 𝐶 𝑛 big-q-Jacobi-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 big-q-Jacobi-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 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)+A_{n-1}C_{n}{\widehat{P}}_{n-1}\!\left(x\right)}}}

Substitution(s)

C n = - a ⁢ c ⁢ q n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ c - 1 ⁢ q n ) ⁢ ( 1 - b ⁢ q n ) ( 1 - a ⁢ b ⁢ q 2 ⁢ n ) ⁢ ( 1 - a ⁢ b ⁢ q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝑎 𝑐 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑐 1 superscript 𝑞 𝑛 1 𝑏 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 2 𝑛 1 𝑎 𝑏 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=-acq^{n+1}\frac{(1-q^{n})(1-% abc^{-1}q^{n})(1-bq^{n})}{(1-abq^{2n})(1-abq^{2n+1})}}}} &
A n = ( 1 - a ⁢ q n + 1 ) ⁢ ( 1 - a ⁢ b ⁢ q n + 1 ) ⁢ ( 1 - c ⁢ q n + 1 ) ( 1 - a ⁢ b ⁢ q 2 ⁢ n + 1 ) ⁢ ( 1 - a ⁢ b ⁢ q 2 ⁢ n + 2 ) subscript 𝐴 𝑛 1 𝑎 superscript 𝑞 𝑛 1 1 𝑎 𝑏 superscript 𝑞 𝑛 1 1 𝑐 superscript 𝑞 𝑛 1 1 𝑎 𝑏 superscript 𝑞 2 𝑛 1 1 𝑎 𝑏 superscript 𝑞 2 𝑛 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-aq^{n+1})(1-abq^{n+1% })(1-cq^{n+1})}{(1-abq^{2n+1})(1-abq^{2n+2})}}}}


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}}} -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:monicbigqJacobi

Bibliography

Equation in Section 14.5 of KLS.

URL links

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