Formula:KLS:14.10:09

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P ~ n ( n , β ) ⁡ ( x | q ) := P ~ n ( α , β ) ⁡ ( x | q ) = ( q ; q ) n ( q α + 1 ; q ) n ⁢ P n ( α , β ) ⁡ ( x | q ) assign continuous-q-Jacobi-polynomial-P-tilde 𝑛 𝛽 𝑛 𝑥 𝑞 continuous-q-Jacobi-polynomial-P-tilde 𝛼 𝛽 𝑛 𝑥 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 continuous-q-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{P}}^{(n,\beta)}_{n}\!\left(x% |q\right):={\tilde{P}}^{(\alpha,\beta)}_{n}\!\left(x|q\right)=\frac{\left(q;q% \right)_{n}}{\left(q^{\alpha+1};q\right)_{n}}P^{(\alpha,\beta)}_{n}\!\left(x|q% \right)}}}

Proof

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Symbols List

P ~ n ( α , β ) subscript superscript ~ 𝑃 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle{\tilde{P}}^{(\alpha,\beta)}_{n}}}}  : normalized continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Jacobi polynomial P ~ ~ 𝑃 {\displaystyle{\displaystyle{\displaystyle{\tilde{P}}}}}  : http://drmf.wmflabs.org/wiki/Definition:normctsqJacobiPtilde
( 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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