Formula:KLS:14.10:69: Difference between revisions

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Latest revision as of 08:37, 22 December 2019


C n ⁡ ( x ; q α + 1 2 | q ) = ( q 2 ⁢ α + 1 ; q ) n ( q α + 1 ; q ) n ⁢ q ( 1 2 ⁢ α + 1 4 ) ⁢ n ⁢ P n ( α , α ) ⁡ ( x | q ) continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 superscript 𝑞 𝛼 1 2 𝑞 q-Pochhammer-symbol superscript 𝑞 2 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 superscript 𝑞 1 2 𝛼 1 4 𝑛 continuous-q-Jacobi-polynomial-P 𝛼 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;q^{\alpha+\frac{1}{2% }}\,|\,q\right)=\frac{\left(q^{2\alpha+1};q\right)_{n}}{\left(q^{\alpha+1};q% \right)_{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}P^{(\alpha,\alpha)}_{n}\!\left% (x|q\right)}}}

Proof

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Symbols List

C n subscript 𝐶 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -ultraspherical/Rogers polynomial : http://dlmf.nist.gov/18.28#E13
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
P n ( α , β ) subscript superscript 𝑃 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi

Bibliography

Equation in Section 14.10 of KLS.

URL links

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