Formula:KLS:14.20:04

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- x ⁢ p n ⁡ ( x ; a ; q ) = A n ⁢ p n + 1 ⁡ ( x ; a ; q ) - ( A n + C n ) ⁢ p n ⁡ ( x ; a ; q ) + C n ⁢ p n - 1 ⁡ ( x ; a ; q ) 𝑥 little-q-Laguerre-Wall-polynomial-p 𝑛 𝑥 𝑎 𝑞 subscript 𝐴 𝑛 little-q-Laguerre-Wall-polynomial-p 𝑛 1 𝑥 𝑎 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 little-q-Laguerre-Wall-polynomial-p 𝑛 𝑥 𝑎 𝑞 subscript 𝐶 𝑛 little-q-Laguerre-Wall-polynomial-p 𝑛 1 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle-xp_{n}\!\left(x;a;q\right)=A_{n}p_{% n+1}\!\left(x;a;q\right)-\left(A_{n}+C_{n}\right)p_{n}\!\left(x;a;q\right)+C_{% n}p_{n-1}\!\left(x;a;q\right)}}}

Substitution(s)

C n = a ⁢ q n ⁢ ( 1 - q n ) subscript 𝐶 𝑛 𝑎 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}=aq^{n}(1-q^{n})}}} &
A n = q n ⁢ ( 1 - a ⁢ q n + 1 ) subscript 𝐴 𝑛 superscript 𝑞 𝑛 1 𝑎 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=q^{n}(1-aq^{n+1})}}}


Proof

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Symbols List

& : logical and
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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