Formula:KLS:14.03:15

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( 1 - q ) 2 ⁢ D q ⁢ [ w ~ ⁢ ( x ; a ⁢ q 1 2 , b ⁢ q 1 2 ⁢ c ⁢ q 1 2 | q ) ⁢ D q ⁢ y ⁢ ( x ) ] + 4 ⁢ q - n + 1 ⁢ ( 1 - q n ) ⁢ w ~ ⁢ ( x ; a , b , c | q ) ⁢ y ⁢ ( x ) = 0 superscript 1 𝑞 2 subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 conditional 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑞 subscript 𝐷 𝑞 𝑦 𝑥 4 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 ~ 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle(1-q)^{2}D_{q}\left[{\tilde{w}}(x;aq% ^{\frac{1}{2}},bq^{\frac{1}{2}}cq^{\frac{1}{2}}|q)D_{q}y(x)\right]{}+4q^{-n+1}% (1-q^{n}){\tilde{w}}(x;a,b,c|q)y(x)=0}}}

Substitution(s)

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

y ⁢ ( x ) = p n ⁡ ( x ; a , b , c | q ) 𝑦 𝑥 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=p_{n}\!\left(x;a,b,c|q\right)}}} &
w ⁢ ( x ) := w ⁢ ( x ; a , b , c | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 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 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}},c{\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)h(x,b)h(x,c)}}}} &
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
p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : continuous dual q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsdualqHahn
( 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 14.3 of KLS.

URL links

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