Formula:KLS:14.20:14: Difference between revisions

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Latest revision as of 08:38, 22 December 2019


𝒟 q - 1 ⁢ [ w ⁢ ( x ; α | q ) ⁢ p n ⁡ ( x ; q α ; q ) ] = 1 - q α q α - 1 ⁢ ( 1 - q ) ⁢ w ⁢ ( x ; α - 1 | q ) ⁢ p n + 1 ⁡ ( x ; q α - 1 ; q ) subscript 𝒟 superscript 𝑞 1 delimited-[] 𝑤 𝑥 conditional 𝛼 𝑞 little-q-Laguerre-Wall-polynomial-p 𝑛 𝑥 superscript 𝑞 𝛼 𝑞 1 superscript 𝑞 𝛼 superscript 𝑞 𝛼 1 1 𝑞 𝑤 𝑥 𝛼 conditional 1 𝑞 little-q-Laguerre-Wall-polynomial-p 𝑛 1 𝑥 superscript 𝑞 𝛼 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q^{-1}}\left[w(x;\alpha% |q)p_{n}\!\left(x;q^{\alpha};q\right)\right]{}=\frac{1-q^{\alpha}}{q^{\alpha-1% }(1-q)}w(x;\alpha-1|q)p_{n+1}\!\left(x;q^{\alpha-1};q\right)}}}

Substitution(s)

w ⁢ ( x ; α | q ) = ( q ⁢ x ; q ) ∞ ⁢ x α 𝑤 𝑥 conditional 𝛼 𝑞 q-Pochhammer-symbol 𝑞 𝑥 𝑞 superscript 𝑥 𝛼 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha|q)=\left(qx;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}}} -Laguerre / Wall polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqLaguerre
( 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 14.20 of KLS.

URL links

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