Product formula for Hermite polynomials

[ksoileau]

I'm looking for a formula which expresses the product of two Hermite polynomials as a linear combination of Hermite polynomials, i.e. $a_{m,n,i}$ verifying
$$
H_m(x)H_n(x)=\sum \limits_{i=0}^{m+n} a_{m,n,i} H_i(x).
$$
for all nonegative $m,n$.

If such a formula is known, I'd be most appreciative of a citation or link describing it.

Thanks for any help!

Kerry M. Soileau

[red...@siu.edu]

Try Google. The first entry in my search gave an answer to your question.

Don

[ksoileau]

On Saturday, January 19, 2013 3:38:03 PM UTC-6, ksoileau wrote:

I already did that and found no answer to my question at any of these links:

http://en.wikipedia.org/wiki/Orthogonal_polynomials
http://mathworld.wolfram.com/HermitePolynomial.html
http://en.wikipedia.org/wiki/Hermite_polynomials

If the answer was so easy to find using Google, why did you take the trouble to write a reply without providing a link? You have a rather eccentric concept of "helpfulness."

Thanks anyway!

[red...@siu.edu]

www.math.niu.edu/~rusin/known-math/99/prod-hermite

L. Carlitz, The product of certain polynomial analogues to the Hermite polynomials, Amer. Math. Monthly 64(1957), 723-725

Hope these help.

Don

[Roland Franzius]

Am 19.01.2013 22:38, schrieb ksoileau:
> I'm looking for a formula which expresses the product of two Hermite polynomials as a linear combination of Hermite polynomials, i.e. $a_{m,n,i}$ verifying
> $$
> H_m(x)H_n(x)=\sum \limits_{i=0}^{m+n} a_{m,n,i} H_i(x).
> $$
> for all nonegative $m,n$.
>
> If such a formula is known, I'd be most appreciative of a citation or link describing it.

Since the functions {H_n(x),n=0,1,2...} form a system of orthogonal
polynomials with respect to the real inner product
< a, b > = int_R dx exp(-x^2) a(x) b(x)

all one needs is the decomposition of the product H_n*H_m .

The relevant scalar product formula is

< H_n * H_m , H_k > =
sqrt(pi) 2^((n+m+k)/2) n!m!k!/((n+m-k)/2)!/((n+k-m)/2!/((m+k-n)/2)!

(n+m+k) even

see eg Gradshteyn/Rhyzik or Erdelyi

--

Roland Franzius

[ksoileau]

Roland,
Many thanks! That was exactly what I needed.
Best,
Kerry Soileau

[AP]

an other link
http://thesis.library.caltech.edu/1861/1/wuan_thesis.pdf

lemme 2.2 with the proof