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

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Roland Franzius

Posts: 586
Registered: 12/7/04
Re: Product formula for Hermite polynomials
Posted: Jan 20, 2013 7:23 PM
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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

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