Formula:KLS:09.12:31

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H 2 ⁢ m μ ⁡ ( x ) := c ⁢ o ⁢ n ⁢ s ⁢ t ⁢ L m μ - 1 2 ⁡ ( x 2 ) , H 2 ⁢ m + 1 μ ⁡ ( x ) : : assign generalized-Hermite-polynomial-H 𝜇 2 𝑚 𝑥 c 𝑜 𝑛 𝑠 𝑡 generalized-Laguerre-polynomial-L 𝜇 1 2 𝑚 superscript 𝑥 2 generalized-Hermite-polynomial-H 𝜇 2 𝑚 1 𝑥 absent {\displaystyle{\displaystyle{\displaystyle H^{\mu}_{2m}\left(x\right):=\mathrm% {c}onstL^{\mu-\frac{1}{2}}_{m}\left(x^{2}\right),\qquad H^{\mu}_{2m+1}\left(x% \right):}}}

Proof

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Symbols List

H n μ superscript subscript 𝐻 𝑛 𝜇 {\displaystyle{\displaystyle{\displaystyle H_{n}^{\mu}}}}  : generalized Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:GenHermite
L n ( α ) superscript subscript 𝐿 𝑛 𝛼 {\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}  : Laguerre (or generalized Laguerre) polynomial : http://dlmf.nist.gov/18.3#T1.t1.r27

Bibliography

Equation in Section 9.12 of KLS.

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