Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Product formula for Hermite polynomials
Replies: 5   Last Post: Jan 24, 2013 3:50 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Roland Franzius

Posts: 417
Registered: 12/7/04
Re: Product formula for Hermite polynomials
Posted: Jan 20, 2013 7:23 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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




Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.