From: Mike Stay <metaweta@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Fri, October 30, 2009 12:21:06 PM Subject: [math-fun] Integral of a Gaussian times a Hermite polynomial Mathworld's online integrator seems to be able to handle \int exp(-x^2/2) H_n(x) dx for any particular value of n, but fails for the general case; what's the general formula? -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com ________________________________ Refer to the Wiki page on Hermite polynomials. There are two conventions, one based on weight function exp(-x^2/2), the other based on exp(-x^2). In the former case, we have int(exp(-x^2/2) H[n](x), x=-inf..inf) = 0 for n>0, as this is the orthogonality relation. So I assume you are using the latter case. We have the generating function exp(2xt - t^2) = sum(H[n](x) t^n/n!, n=0..inf). Let I[n] = int(exp(-x^2/2) H[n](x), x=-inf..inf). Then sum(exp(-x^2/2) H[n](x) t^n/n!, n=0..inf) = exp(-x^2/2 + 2xt - t^2) = exp(-(x-2t)^2/2 + t^2). sum(I[n] t^n/n!, n=0..inf) = int(exp(-(x-2t)^2/2 + t^2), x=-inf..inf) = sqrt(2 pi) exp(t^2) = sqrt(2 pi) sum(t^(2n)/n!, n=0..inf). Thus, I[n] = sqrt(2 pi) n!/(n/2)! for n even, I[n] = 0 for n odd. -- Gene