Formula:KLS:14.26:08

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( 1 - q ) 2 D q [ w ~ ( x | q ) D q y ( x ) ] + 4 q - n + 1 ( 1 - q n ) w ~ ( x | q ) y ( x ) = 0 superscript 1 𝑞 2 subscript 𝐷 𝑞 delimited-[] ~ 𝑤 conditional 𝑥 𝑞 subscript 𝐷 𝑞 𝑦 𝑥 4 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 ~ 𝑤 conditional 𝑥 𝑞 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle(1-q)^{2}D_{q}\left[{\tilde{w}}(x|q)% D_{q}y(x)\right]+4q^{-n+1}(1-q^{n}){\tilde{w}}(x|q)y(x)=0}}}

Substitution(s)

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

y ( x ) = H n ( x | q ) 𝑦 𝑥 continuous-q-Hermite-polynomial-H 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=H_{n}\!\left(x\,|\,q\right)}}} &
w ( x | q ) = | ( e 2 i θ ; q ) | 2 = h ( x , 1 ) h ( x , - 1 ) h ( x , q 1 2 ) h ( x , - q 1 2 ) 𝑤 conditional 𝑥 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 2 𝑥 1 𝑥 1 𝑥 superscript 𝑞 1 2 𝑥 superscript 𝑞 1 2 {\displaystyle{\displaystyle{\displaystyle w(x|q)=\left|\left({\mathrm{e}^{2% \mathrm{i}\theta}};q\right)_{\infty}\right|^{2}=h(x,1)h(x,-1)h(x,q^{\frac{1}{2% }})h(x,-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 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
H n subscript 𝐻 𝑛 {\displaystyle{\displaystyle{\displaystyle H_{n}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqHermite
( 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 14.26 of KLS.

URL links

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