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Similar to the question Legendre Polynomials Triple Product, I would like to ask whether there are any explicit formulas for the inner product of the Hermite polynomial triple product \begin{align} \int_{-\infty}^{\infty} P_k(x)\,P_l(x)\, P_m(x)\ w(x)\;dx \end{align} where $P_{k,l,m}$ are Hermite Polynomials and $w(x)=\exp(-\frac{x^2}{2})$ is the corresponding weight function.

Can one generalize the application of the 3j symbol for Legendre polynomials (http://theoretical-physics.net/dev/src/math/spherical-harmonics.html#adams) towards other orthogonal polynomials, or if not, why can it be applied to the Legendre polynomial (i.e. is there an intuitive explanation)?

Otherwise, any ideas about how I could calculate the integrals for different values of k,l and m efficiently? Are there tables/references where I can find the values?

Thanks in advance!

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1 Answer 1

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Yes, such formulas exist (they are related to the so-called "linearization formulas"), and can be found in the literature. For instance, for the standard Hermite polynomials $H_k(x)$, the corresponding formula is: $$ \int_{-\infty}^\infty H_\ell(x) \, H_m(x) \, H_n(x) \, e^{-x^2} dx = \frac{2^{(\ell+m+n)/2}\ell! m! n! \sqrt{\pi}}{(\frac{\ell+m-n}{2})!(\frac{m+n-\ell}{2})!(\frac{n+\ell-m}{2})!} $$ if $\ell+m+n$ is even and the sum of any two of $\ell,m,n$ is not less than the third, or zero otherwise. This is equation (6.8.3) in the book "Special Functions", by Andrews-Askey-Roy.

Notice that your weight function is slightly different (it's a scaled version, corresponding to the "probabilists' Hermite polynomials"), but an easy change of variables will give the result that you want.

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