Formula:KLS:14.01:23

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δ q ⁢ p n ⁡ ( x ; a , b , c , d | q ) = - q - 1 2 ⁢ n ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ p n - 1 ⁡ ( x ; a ⁢ q 1 2 , b ⁢ q 1 2 , c ⁢ q 1 2 , d ⁢ q 1 2 | q ) subscript 𝛿 𝑞 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 superscript 𝑞 1 2 𝑛 1 superscript 𝑞 𝑛 1 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 imaginary-unit 𝜃 imaginary-unit 𝜃 Askey-Wilson-polynomial-p 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑑 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}p_{n}\!\left(x;a,b,c,d\,|% \,q\right)=-q^{-\frac{1}{2}n}(1-q^{n})(1-abcdq^{n-1})({\mathrm{e}^{\mathrm{i}% \theta}}-{\mathrm{e}^{-\mathrm{i}\theta}}){}p_{n-1}\!\left(x;aq^{\frac{1}{2}},% bq^{\frac{1}{2}},cq^{\frac{1}{2}},dq^{\frac{1}{2}}\,|\,q\right)}}}

Proof

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Symbols List

p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : Askey-Wilson polynomial : http://dlmf.nist.gov/18.28#E1
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 14.1 of KLS.

URL links

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