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...
