Formula:KLS:14.18:09

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2 ⁢ x ⁢ H n ⁡ ( x ; a | q ) = H n + 1 ⁡ ( x ; a | q ) + a ⁢ q n ⁢ H n ⁡ ( x ; a | q ) + ( 1 - q n ) ⁢ H n - 1 ⁡ ( x ; a | q ) 2 𝑥 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 continuous-big-q-Hermite-polynomial-H 𝑛 1 𝑥 𝑎 𝑞 𝑎 superscript 𝑞 𝑛 continuous-big-q-Hermite-polynomial-H 𝑛 𝑥 𝑎 𝑞 1 superscript 𝑞 𝑛 continuous-big-q-Hermite-polynomial-H 𝑛 1 𝑥 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle 2xH_{n}\!\left(x;a|q\right)=H_{n+1}% \!\left(x;a|q\right)+aq^{n}H_{n}\!\left(x;a|q\right)+(1-q^{n})H_{n-1}\!\left(x% ;a|q\right)}}}

Proof

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Symbols List

H n subscript 𝐻 𝑛 {\displaystyle{\displaystyle{\displaystyle H_{n}}}}  : continuous big q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite

Bibliography

Equation in Section 14.18 of KLS.

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