## Derivative recurrence relation for Hermite polynomials

Posted by peeterjoot on January 3, 2012

# Motivation.

For a QM problem I had need of a recurrence relation for Hermite polynomials. I found it in [1], but thought I’d try to derive the relation myself.

# Guts

The starting point I’ll use is the Rodgigues’ formula definition of the Hermite polynomials

Let’s write , and take the derivative of

So we have the rather simple end result

# References

[1] M. Abramowitz and I.A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55. Dover publications, 1964.

Advertisements

## Mike said

Very nice derivation! For completeness, I think you’re missing a factor of (-1)^n from the start of the main calculation.

## peeterjoot said

Good eyes Mike. Corrected.