Formula:KLS:14.10:61

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\qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ β 1 2 ⁢ e i ⁢ θ - β 1 2 ⁢ q 1 2 ⁢ e i ⁢ θ - β ⁢ q 1 2 ⁢ q ⁢ e - i ⁢ θ ⁢ t ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ β 1 2 ⁢ q 1 2 ⁢ e - i ⁢ θ - β 1 2 ⁢ e - i ⁢ θ - β ⁢ q 1 2 ⁢ q ⁢ e i ⁢ θ ⁢ t = ∑ n = 0 ∞ ( - β , β ⁢ q 1 2 ; q ) n ( β 2 , - β ⁢ q 1 2 ; q ) n ⁢ C n ⁡ ( x ; β | q ) ⁢ t n \qHyperrphis 21 @ @ superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 𝛽 superscript 𝑞 1 2 𝑞 imaginary-unit 𝜃 𝑡 \qHyperrphis 21 @ @ superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 imaginary-unit 𝜃 𝛽 superscript 𝑞 1 2 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝛽 𝛽 superscript 𝑞 1 2 𝑞 𝑛 q-Pochhammer-symbol superscript 𝛽 2 𝛽 superscript 𝑞 1 2 𝑞 𝑛 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{\beta^{\frac{1}% {2}}{\mathrm{e}^{\mathrm{i}\theta}}-\beta^{\frac{1}{2}}q^{\frac{1}{2}}{\mathrm% {e}^{\mathrm{i}\theta}}}{-\beta q^{\frac{1}{2}}}{q}{{\mathrm{e}^{-\mathrm{i}% \theta}}t}\ \qHyperrphis{2}{1}@@{\beta^{\frac{1}{2}}q^{\frac{1}{2}}{\mathrm{e}% ^{-\mathrm{i}\theta}}-\beta^{\frac{1}{2}}{\mathrm{e}^{-\mathrm{i}\theta}}}{-% \beta q^{\frac{1}{2}}}{q}{{\mathrm{e}^{\mathrm{i}\theta}}t}{}=\sum_{n=0}^{% \infty}\frac{\left(-\beta,\beta q^{\frac{1}{2}};q\right)_{n}}{\left(\beta^{2},% -\beta q^{\frac{1}{2}};q\right)_{n}}C_{n}\!\left(x;\beta\,|\,q\right)t^{n}}}}

Substitution(s)

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Proof

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Symbols List

ϕ s r subscript subscript italic-ϕ 𝑠 𝑟 {\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}  : basic hypergeometric (or q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -hypergeometric) function : http://dlmf.nist.gov/17.4#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
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( 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
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
cos cos {\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}  : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.10 of KLS.

URL links

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