Formula:KLS:14.20:11

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p n ⁡ ( x ; a ; q ) - p n ⁡ ( q ⁢ x ; a ; q ) = - q - n + 1 ⁢ ( 1 - q n ) 1 - a ⁢ q ⁢ x ⁢ p n - 1 ⁡ ( x ; a ⁢ q ; q ) little-q-Laguerre-Wall-polynomial-p 𝑛 𝑥 𝑎 𝑞 little-q-Laguerre-Wall-polynomial-p 𝑛 𝑞 𝑥 𝑎 𝑞 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑎 𝑞 𝑥 little-q-Laguerre-Wall-polynomial-p 𝑛 1 𝑥 𝑎 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a;q\right)-p_{n}\!% \left(qx;a;q\right)=-\frac{q^{-n+1}(1-q^{n})}{1-aq}xp_{n-1}\!\left(x;aq;q% \right)}}}

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

Bibliography

Equation in Section 14.20 of KLS.

URL links

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