Formula:KLS:14.05:03: Difference between revisions

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( x - 1 ) ⁢ P n ⁡ ( x ; a , b , c ; q ) = A n ⁢ P n + 1 ⁡ ( x ; a , b , c ; q ) - ( A n + C n ) ⁢ P n ⁡ ( x ; a , b , c ; q ) + C n ⁢ P n - 1 ⁡ ( x ; a , b , c ; q ) 𝑥 1 big-q-Jacobi-polynomial-P 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 subscript 𝐴 𝑛 big-q-Jacobi-polynomial-P 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 big-q-Jacobi-polynomial-P 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 subscript 𝐶 𝑛 big-q-Jacobi-polynomial-P 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle(x-1)P_{n}\!\left(x;a,b,c;q\right)=A% _{n}P_{n+1}\!\left(x;a,b,c;q\right)-\left(A_{n}+C_{n}\right)P_{n}\!\left(x;a,b% ,c;q\right){}+C_{n}P_{n-1}\!\left(x;a,b,c;q\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 P_{n}}}}  : big q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:bigqJacobi

Bibliography

Equation in Section 14.5 of KLS.

URL links

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