Formula:KLS:14.05:21

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lim a → 0 ⁡ p ~ n ⁡ ( 1 2 ⁢ a - 1 ⁢ x ; a , a - 1 ⁢ α ⁢ q , a - 1 ⁢ γ ⁢ q , a ⁢ β ⁢ γ - 1 | q ) = P n ⁡ ( x ; α , β , γ ; q ) subscript → 𝑎 0 Askey-Wilson-polynomial-normalized-p-tilde 𝑛 1 2 superscript 𝑎 1 𝑥 𝑎 superscript 𝑎 1 𝛼 𝑞 superscript 𝑎 1 𝛾 𝑞 𝑎 𝛽 superscript 𝛾 1 𝑞 big-q-Jacobi-polynomial-P 𝑛 𝑥 𝛼 𝛽 𝛾 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow 0}{\tilde{p}}_{n}% \!\left(\textstyle\frac{1}{2}a^{-1}x;a,a^{-1}\alpha q,a^{-1}\gamma q,a\beta% \gamma^{-1}\,|\,q\right)=P_{n}\!\left(x;\alpha,\beta,\gamma;q\right)}}}

Proof

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Symbols List

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
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.

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