Formula:KLS:14.04:20

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w ~ ⁢ ( x ; a , b , c , d ; q ) ⁢ p n ⁡ ( x ; a , b , c , d ; q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n ⁢ ( n - 1 ) ⁢ ( D q ) n ⁢ [ w ~ ⁢ ( x ; a ⁢ q 1 2 ⁢ n , b ⁢ q 1 2 ⁢ n , c ⁢ q 1 2 ⁢ n , d ⁢ q 1 2 ⁢ n ; q ) ] ~ 𝑤 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 continuous-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 𝑛 𝑛 1 superscript subscript 𝐷 𝑞 𝑛 delimited-[] ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 𝑛 𝑏 superscript 𝑞 1 2 𝑛 𝑐 superscript 𝑞 1 2 𝑛 𝑑 superscript 𝑞 1 2 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c,d;q)p_{n}\!\left% (x;a,b,c,d;q\right){}=\left(\frac{q-1}{2}\right)^{n}q^{\frac{1}{4}n(n-1)}\left% (D_{q}\right)^{n}\left[{\tilde{w}}(x;aq^{\frac{1}{2}n},bq^{\frac{1}{2}n},cq^{% \frac{1}{2}n},dq^{\frac{1}{2}n};q)\right]}}}

Substitution(s)

w ~ ⁢ ( x ; a , b , c , d ; q ) := w ⁢ ( x ; a , b , c , d ; q ) 1 - x 2 assign ~ 𝑤 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 𝑤 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c,d;q):=\frac{w(x;% a,b,c,d;q)}{\sqrt{1-x^{2}}}}}} &

w ⁢ ( x ) := w ⁢ ( x ; a , b , c , d ; q ) = | ( e 2 ⁢ i ⁢ ( θ + ϕ ) ; q ) ∞ ( a ⁢ e i ⁢ ( θ + 2 ⁢ ϕ ) , b ⁢ e i ⁢ ( θ + 2 ⁢ ϕ ) ⁢ c ⁢ e i ⁢ θ , d ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ⁢ e i ⁢ ϕ ) ⁢ h ⁢ ( x , b ⁢ e i ⁢ ϕ ) ⁢ h ⁢ ( x , c ⁢ e - i ⁢ ϕ ) ⁢ h ⁢ ( x , d ⁢ e - i ⁢ ϕ ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 italic-ϕ 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 2 italic-ϕ 𝑏 imaginary-unit 𝜃 2 italic-ϕ 𝑐 imaginary-unit 𝜃 𝑑 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 𝑎 imaginary-unit italic-ϕ ℎ 𝑥 𝑏 imaginary-unit italic-ϕ ℎ 𝑥 𝑐 imaginary-unit italic-ϕ ℎ 𝑥 𝑑 imaginary-unit italic-ϕ {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c,d;q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}{\left(a{% \mathrm{e}^{\mathrm{i}(\theta+2\phi)}},b{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}% }c{\mathrm{e}^{\mathrm{i}\theta}},d{\mathrm{e}^{\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,a{\mathrm{e}^{\mathrm{i}\phi}})h(x,b{\mathrm{e}^{\mathrm{i}\phi}})h(x,% c{\mathrm{e}^{-\mathrm{i}\phi}})h(x,d{\mathrm{e}^{-\mathrm{i}\phi}})}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ ( θ + ϕ ) , α ⁢ e - i ⁢ ( θ + ϕ ) ; 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 italic-ϕ 𝛼 imaginary-unit 𝜃 italic-ϕ 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+\phi)}},\alpha{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}};q% \right)_{\infty}}}} &

x = cos ⁡ ( θ + ϕ ) 𝑥 𝜃 italic-ϕ {\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}


Proof

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Symbols List

& : logical and
p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqHahn
( 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 14.4 of KLS.

URL links

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