[Math] Hermite Polynomials Triple Product

integrationinterpolationorthogonal-polynomials

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!

Best Answer

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.

Related Solutions

Legendre Polynomials Triple Product – Integration and Orthogonal Polynomials

I think the best way to approach this is as follows, note that \begin{align} (x^2 -1 ) = \frac{P_2 - 2}{3} \end{align} You can then use the following definition \begin{align} P_kP_l = \sum_{m=|k-l|}^{k+l} \begin{pmatrix} k & l & m \\ 0 & 0 & 0 \end{pmatrix}^2 (2m+1)P_m \end{align} This allows the integral to be written as follows \begin{align} \int_{-1}^{1} (x^2-1)^3P_iP_jP_k \; dx &= \int_{-1}^{1} \frac{1}{9}\left(P_2^3 + . . .-8 \right) P_i P_j P_k \; dx \end{align} The most difficult term to deal with is the $ P_2^3 P_i P_j P_k$ \begin{align} P_2^3 P_i P_j P_k &= \sum_{m=0}^{4} \begin{pmatrix} 2 & 2 & m \\ 0 & 0 & 0 \end{pmatrix}^2 (2m+1)P_m P_2 P_i P_j P_k \\ &= \sum_{m=0}^{4} \begin{pmatrix} 2 & 2 & m \\ 0 & 0 & 0 \end{pmatrix}^2 (2m+1) \sum_{n=|m-2|}^{m+2} \begin{pmatrix} 2 & m & n \\ 0 & 0 & 0 \end{pmatrix}^2 (2n+1)P_n P_i P_j P_k \\ &= \sum_{m=0}^{4} \begin{pmatrix} 2 & 2 & m \\ 0 & 0 & 0 \end{pmatrix}^2 (2m+1) \sum_{n=|m-2|}^{m+2} \begin{pmatrix} 2 & m & n \\ 0 & 0 & 0 \end{pmatrix}^2 (2n+1) \sum_{l=|n-i|}^{n+i} \begin{pmatrix} n & i & l \\ 0 & 0 & 0 \end{pmatrix}^2 (2l+1) P_l P_j P_k \end{align} Which can then make use of the usual triple integral. All other terms can be solved for in a similar manner.

[Math] Calculate orthonormal basis using Gram-Schmidt

The polynomials obtained from (1) are the normalized Legendre polynomials. For all three cases, it is possible to calculate them by hand, but (especially for (3)) the calculation can be quite cumbersome. So, I've used the following Mathematica code for the Gram-Schmidt (including normalization) procedure:

Clear[f, g, ip];
ip[f_, g_] := Integrate[f*g*1, {x, -1, 1}];
proj[g_, f_] := g*ip[f, g]/ip[g, g];
normalized[f_] := f/Sqrt[ip[f, f]];
gs[fs_List] := Module[{u}, normalized /@ 
Table[u[k] = fs[[k]] - Sum[proj[u[j], fs[[k]]], {j, 1, k - 1}],
{k, 1, Length[fs]}]];

lps = Expand[gs[Table[x^k, {k, 0, 4}]]]

This is the code for the first part of the question - if you want to do it for (2) and (3), just replace the part in the second line after f * g with the appropriate weight function and the correct interval.

Best Answer

Related Solutions

Legendre Polynomials Triple Product – Integration and Orthogonal Polynomials

[Math] Calculate orthonormal basis using Gram-Schmidt

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