Formula:KLS:14.12:17

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w ⁢ ( x ; α , β | q ) ⁢ p n ⁡ ( x ; q α , q β ; q ) = q n ⁢ α + \binomial ⁢ n ⁢ 2 ⁢ ( 1 - q ) n ( q α + 1 ; q ) n ⁢ ( 𝒟 q - 1 ) n ⁢ [ w ⁢ ( x ; α + n , β + n | q ) ] 𝑤 𝑥 𝛼 conditional 𝛽 𝑞 little-q-Jacobi-polynomial-p 𝑛 𝑥 superscript 𝑞 𝛼 superscript 𝑞 𝛽 𝑞 superscript 𝑞 𝑛 𝛼 \binomial 𝑛 2 superscript 1 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 superscript subscript 𝒟 superscript 𝑞 1 𝑛 delimited-[] 𝑤 𝑥 𝛼 𝑛 𝛽 conditional 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta|q)p_{n}\!\left(x;q% ^{\alpha},q^{\beta};q\right){}=\frac{q^{n\alpha+\binomial{n}{2}}(1-q)^{n}}{% \left(q^{\alpha+1};q\right)_{n}}\left(\mathcal{D}_{q^{-1}}\right)^{n}\left[w(x% ;\alpha+n,\beta+n|q)\right]}}}

Substitution(s)

w ⁢ ( x ; α , β | q ) = ( q ⁢ x ; q ) ∞ ( q β + 1 ⁢ x ; q ) ∞ ⁢ x α 𝑤 𝑥 𝛼 conditional 𝛽 𝑞 q-Pochhammer-symbol 𝑞 𝑥 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛽 1 𝑥 𝑞 superscript 𝑥 𝛼 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta|q)=\frac{\left(qx;% q\right)_{\infty}}{\left(q^{\beta+1}x;q\right)_{\infty}}x^{\alpha}}}}


Proof

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Symbols List

p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : little q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqJacobi
( n k ) binomial 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}  : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
( 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

Bibliography

Equation in Section 14.12 of KLS.

URL links

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