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Define Hermite Polynomials to Sturm-Liouville Form?

Expert Solution

What is Hermite Polynomials?

For a Hermite Polynomial, we begin with the differential equation
$y^{''} - 2 x y^{'} + 2 n y = 0$

First, we need to arrange the differential equation so it can be written in the form:

$(p (x) y$ ^')'+(q(x)+λr(x))y=0

For the Hermite differential equation, we use $r (x) = e^{- x^{2}}$ to get

$e^{- x^{2}} y^{''} - 2 x e^{- x^{2}} y^{'} + 2 n e^{- x^{2}} y = 0$

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