
Re: Product formula for Hermite polynomials
Posted:
Jan 20, 2013 7:23 PM


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+mk)/2)!/((n+km)/2!/((m+kn)/2)!
(n+m+k) even
see eg Gradshteyn/Rhyzik or Erdelyi

Roland Franzius

