MHB Relation between Hermite and associated Laguerre

[Suvadip]

Please help me in in proving the relation between H_2n(x) and L_n^(-1/2)(x²) where H_n(x) is the Hermite polynomial and L_n(-1/2)(x) is associated Laguerre polynomial.

 

[DreamWeaver]

Hi Suvadip! :D

Are you familiar with "n-th derivative" definitions of the Hermite and Laguerre polynomials? The standard definition for the Laguerre polynomials is

$$\mathcal{L}_n(x)=\frac{e^x}{n!}\,\frac{d^n}{dx^n}[e^{-x}x^n]$$

although you do also occasionally come across the alternate form

$$ \mathcal{L}_n(x)=e^x\,\frac{d^n}{dx^n}[e^{-x}x^n]$$

The latter definition omits the scale factor $$1/{n!}\,$$ and so modifies the recurrence relations...On the other hand, the Hermite polynomials have the analogous definition:

$$\mathcal{H}_n(x)=(-1)^n\,e^{x^2}\,\frac{d^n}{dx^n}[e^{-x^2}]$$

You might try differentiating all 3 definitions - it's good practice! - and then see what happens... ;)

Also, you might consider expressing the terms to be n-differentiated as a power series, differentiating, multiplying back with the remainder of the function (in each respective function definition), and then compare the coefficients for all three results...Bets of luck! :D

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Think fractional integration, that'd be my guess...