Relation between Hermite and associated Laguerre

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    Laguerre Relation
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SUMMARY

The expression relating Hermite polynomials and associated Laguerre polynomials is established as $$H_{2n}(x)=(-1)^n2^{2n}n!L_n^{-\frac{1}{2}}(x^2)$$. Here, $$H_n$$ denotes the Hermite polynomial, while $$L_n^{-\frac{1}{2}}$$ refers to the associated Laguerre polynomial. The discussion emphasizes the importance of demonstrating prior effort in mathematical proofs to facilitate effective assistance from others.

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Suvadip
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Please help me in proving the following expression

$$H_{2n}(x)=(-1)^n2^{2n}n!L_n^{-\frac{1}{2}}(x^2)$$

where $$H_n$$ is the Hermite polynomial and $$L_n^{-\frac{1}{2}}$$ is the associated Laguerre polynomial.
 
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Hello suvadip,

By now you should know we expect some effort to be given, such as the work you have tried, or the thoughts you have on how to proceed, for the reasons I have already given. You may already have tried something, and one of our helpers may give you information you already know, and this would be a waste of the helper's time, which is valuable.

If you simply have no idea how to begin, then you should state this, and ask for a hint to begin.
 

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