Relation between Hermite and associated Laguerre

Member
Please help me in in proving the relation between H2n(x) and Ln(-1/2)(x2) where Hn(x) is the Hermite polynomial and Ln(-1/2)(x) is associated Laguerre polynomial.

DreamWeaver

Well-known member

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

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

although you do also occasionally come across the alternate form

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

The latter definition omits the scale factor $$\displaystyle 1/{n!}\,$$ and so modifies the recurrence relations...

On the other hand, the Hermite polynomials have the analogous definition:

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

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!

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