Formula:KLS:14.12:15

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π’Ÿ q - 1 ⁒ [ w ⁒ ( x ; Ξ± , Ξ² | q ) ⁒ p n ⁑ ( x ; q Ξ± , q Ξ² ; q ) ] = 1 - q Ξ± q Ξ± - 1 ⁒ ( 1 - q ) ⁒ w ⁒ ( x ; Ξ± - 1 , Ξ² - 1 | q ) ⁒ p n + 1 ⁑ ( x ; q Ξ± - 1 , q Ξ² - 1 ; q ) subscript π’Ÿ superscript π‘ž 1 delimited-[] 𝑀 π‘₯ 𝛼 conditional 𝛽 π‘ž little-q-Jacobi-polynomial-p 𝑛 π‘₯ superscript π‘ž 𝛼 superscript π‘ž 𝛽 π‘ž 1 superscript π‘ž 𝛼 superscript π‘ž 𝛼 1 1 π‘ž 𝑀 π‘₯ 𝛼 1 𝛽 conditional 1 π‘ž little-q-Jacobi-polynomial-p 𝑛 1 π‘₯ superscript π‘ž 𝛼 1 superscript π‘ž 𝛽 1 π‘ž {\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q^{-1}}\left[w(x;\alpha% ,\beta|q)p_{n}\!\left(x;q^{\alpha},q^{\beta};q\right)\right]{}=\frac{1-q^{% \alpha}}{q^{\alpha-1}(1-q)}w(x;\alpha-1,\beta-1|q)p_{n+1}\!\left(x;q^{\alpha-1% },q^{\beta-1};q\right)}}}

Substitution(s)

w ⁒ ( x ; Ξ± , Ξ² | q ) = ( q ⁒ x ; q ) ∞ ( q Ξ² + 1 ⁒ x ; q ) ∞ ⁒ x Ξ± 𝑀 π‘₯ 𝛼 conditional 𝛽 π‘ž q-Pochhammer-symbol π‘ž π‘₯ π‘ž q-Pochhammer-symbol superscript π‘ž 𝛽 1 π‘₯ π‘ž superscript π‘₯ 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 π‘ž {\displaystyle{\displaystyle{\displaystyle q}}} -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqJacobi
( a ; q ) n subscript π‘Ž π‘ž 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q π‘ž {\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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