If is a Hermite polynomial of degree n, then show that:&nbs

Question

If $H_{2}$ is a Hermite polynomial of degree n, then show that:

$H_{n}^{n}=4n(n=1)H_{n-2}$

22 Mar 2023

Answer :

Word Count : 249

We can prove this using the generating function of Hermite polynomials. The generating function for Hermite polynomials is given by:

$e^(2xt-t^2) = \sum _(n=0)^\infty H_n(x) (t^n/n!)$

Differentiating both sides of this equation n times with respect to t, we get:

$(\vartheta /\vartheta t)^n (e^(2xt-t^2)) = \sum _(k=n)^\infty H_k(x) (k!/[(k-n)!n!]) t^(k-n)$

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