Formula:KLS:14.01:38

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R n ⁡ ( μ ⁢ ( x ) ; α , β , γ , δ | q ) = ( γ ⁢ δ ⁢ q ) 1 2 ⁢ n ⁢ p n ⁡ ( ν ⁢ ( x ) ; γ 1 2 ⁢ δ 1 2 ⁢ q 1 2 , α ⁢ γ - 1 2 ⁢ δ - 1 2 ⁢ q 1 2 , β ⁢ γ - 1 2 ⁢ δ 1 2 ⁢ q 1 2 , γ 1 2 ⁢ δ - 1 2 ⁢ q 1 2 | q ) ( α ⁢ q , β ⁢ δ ⁢ q , γ ⁢ q ; q ) n q-Racah-polynomial-R 𝑛 𝜇 𝑥 𝛼 𝛽 𝛾 𝛿 𝑞 superscript 𝛾 𝛿 𝑞 1 2 𝑛 Askey-Wilson-polynomial-p 𝑛 𝜈 𝑥 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 1 2 𝛼 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 1 2 𝛽 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 1 2 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 1 2 𝑞 q-Pochhammer-symbol 𝛼 𝑞 𝛽 𝛿 𝑞 𝛾 𝑞 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,\beta,% \gamma,\delta\,|\,q\right){}=\frac{(\gamma\delta q)^{\frac{1}{2}n}p_{n}\!\left% (\nu(x);\gamma^{\frac{1}{2}}\delta^{\frac{1}{2}}q^{\frac{1}{2}},\alpha\gamma^{% -\frac{1}{2}}\delta^{-\frac{1}{2}}q^{\frac{1}{2}},\beta\gamma^{-\frac{1}{2}}% \delta^{\frac{1}{2}}q^{\frac{1}{2}},\gamma^{\frac{1}{2}}\delta^{-\frac{1}{2}}q% ^{\frac{1}{2}}\,|\,q\right)}{\left(\alpha q,\beta\delta q,\gamma q;q\right)_{n% }}}}}

Substitution(s)

ν ⁢ ( x ) = 1 2 ⁢ γ 1 2 ⁢ δ 1 2 ⁢ q x + 1 2 + 1 2 ⁢ γ - 1 2 ⁢ δ - 1 2 ⁢ q - x - 1 2 𝜈 𝑥 1 2 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 𝑥 1 2 1 2 superscript 𝛾 1 2 superscript 𝛿 1 2 superscript 𝑞 𝑥 1 2 {\displaystyle{\displaystyle{\displaystyle\nu(x)=\textstyle\frac{1}{2}\gamma^{% \frac{1}{2}}\delta^{\frac{1}{2}}q^{x+\frac{1}{2}}+\textstyle\frac{1}{2}\gamma^% {-\frac{1}{2}}\delta^{-\frac{1}{2}}q^{-x-\frac{1}{2}}}}}


Proof

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Symbols List

R n subscript 𝑅 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Racah polynomial : http://dlmf.nist.gov/18.28#E19
p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : Askey-Wilson polynomial : http://dlmf.nist.gov/18.28#E1
( 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

Bibliography

Equation in Section 14.1 of KLS.

URL links

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