One typically finds the Hermite differential equation in the context of an infinite square well potential and the consequential solution of the Schrödinger equation. However, I will consider this equation is its “raw” mathematical form viz.

First we will consider the more general case, leaving undefined. The second case will consider in a future post , where

PART I:

Let us assume the solution has the form

Now we take the necessary derivatives

where upon substitution yields the following

Introducing the dummy variable and using this and its variants we arrive at

Bringing this under one summation sign…

Since , we therefore require that

or

This is our recurrence relation. If we let we arrive at two linearly independent solutions (one even and one odd) in terms of the fundamental coefficients and which may be written as

and

Thus, our final solution is the following

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