Re: Hermite Polynomials Overlap Integral

[thk (Tom H. Koornwinder)]

to prof. Bruno Coste

Dear Bruno,
On March 10, you posted to OP-SF TALK the following question:

> I am stuck because I would need an analytic formula for a special
> integration :
> 
> int(-infty,+infty, He(p,ax)*He(q,bx)*exp(-x^2/2),dx)
> 
> and the formulas present in the Bateman Manuscript don't really fit.
> 
> I guess this kind of theoretical calculation has already been
> undertaken.

I am sure that the evaluation of your integral is in the literature.
Here I just point out that you can get the answer as a simple sum (if I
computed wel a terminating Gaussian hypergeometric series) by combining the
following formulas from the Higher Transcendental Functions, Vol.2:

10.12(40), 10.13(2) (or (3)) and the orthogonality relations of
Hermite polynomials.

Kind regards,

		Tom Koornwinder