Formula:KLS:14.10:47

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( 1 - q ) 2 ⁢ D q ⁢ [ w ~ ⁢ ( x ; β ⁢ q | q ) ⁢ D q ⁢ y ⁢ ( x ) ] + λ n ⁢ w ~ ⁢ ( x ; β | q ) ⁢ y ⁢ ( x ) = 0 superscript 1 𝑞 2 subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 conditional 𝛽 𝑞 𝑞 subscript 𝐷 𝑞 𝑦 𝑥 subscript 𝜆 𝑛 ~ 𝑤 𝑥 conditional 𝛽 𝑞 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle(1-q)^{2}D_{q}\left[{\tilde{w}}(x;% \beta q|q)D_{q}y(x)\right]+\lambda_{n}{\tilde{w}}(x;\beta|q)y(x)=0}}}

Substitution(s)

λ n = 4 ⁢ q - n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - β 2 ⁢ q n ) subscript 𝜆 𝑛 4 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 superscript 𝛽 2 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lambda_{n}=4q^{-n+1}(1-q^{n})(1-% \beta^{2}q^{n})}}} &

w ~ ⁢ ( x ; β | q ) := w ⁢ ( x ; β | q ) 1 - x 2 assign ~ 𝑤 𝑥 conditional 𝛽 𝑞 𝑤 𝑥 conditional 𝛽 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;\beta|q):=\frac{w(x;% \beta|q)}{\sqrt{1-x^{2}}}}}} &
y ⁢ ( x ) = C n ⁡ ( x ; β | q ) 𝑦 𝑥 continuous-q-ultraspherical-Rogers-polynomial 𝑛 𝑥 𝛽 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=C_{n}\!\left(x;\beta\,|\,q% \right)}}} &
w ⁢ ( x ) := w ⁢ ( x ; β | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( β 1 2 ⁢ e i ⁢ θ , β 1 2 ⁢ q 1 2 ⁢ e i ⁢ θ - β 1 2 ⁢ e i ⁢ θ , - β 1 2 ⁢ q 1 2 ⁢ e i ⁢ θ ; q ) ∞ | 2 = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( β ⁢ e 2 ⁢ i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , β 1 2 ) ⁢ h ⁢ ( x , β 1 2 ⁢ q 1 2 ) ⁢ h ⁢ ( x , - β 1 2 ) ⁢ h ⁢ ( x , - β 1 2 ⁢ q 1 2 ) assign 𝑤 𝑥 𝑤 𝑥 conditional 𝛽 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 imaginary-unit 𝜃 superscript 𝛽 1 2 superscript 𝑞 1 2 imaginary-unit 𝜃 𝑞 2 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝛽 2 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝛽 1 2 ℎ 𝑥 superscript 𝛽 1 2 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝛽 1 2 ℎ 𝑥 superscript 𝛽 1 2 superscript 𝑞 1 2 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;\beta|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(\beta^{\frac{1% }{2}}{\mathrm{e}^{\mathrm{i}\theta}},\beta^{\frac{1}{2}}q^{\frac{1}{2}}{% \mathrm{e}^{\mathrm{i}\theta}}-\beta^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}% \theta}},-\beta^{\frac{1}{2}}q^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}\theta}};q% \right)_{\infty}}\right|^{2}=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}% };q\right)_{\infty}}{\left(\beta{\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{% \infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}% })}{h(x,\beta^{\frac{1}{2}})h(x,\beta^{\frac{1}{2}}q^{\frac{1}{2}})h(x,-\beta^% {\frac{1}{2}})h(x,-\beta^{\frac{1}{2}}q^{\frac{1}{2}})}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

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


Proof

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Symbols List

& : logical and
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
( 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
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
Π Π {\displaystyle{\displaystyle{\displaystyle\Pi}}}  : product : http://drmf.wmflabs.org/wiki/Definition:prod
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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