How do I evaluate a symbolic integral involving Hermite polynomials?

Question

I want to test a difficult integral : Integral on all reals of some complicated function involving the Hermitian polynomials, exponentials, squares, factorials, and being general considering any Hermitian polynomials. I know the answer (it's ridiculously short compared with the statement) and just wanted to check. But why Mathematica doesn't calculate my integral ? (I'm quite new on the system and don't understand what causes the trouble) :

Integrate[(HermiteH[n, x/a]*Exp[-x^2/(2 a^2)]/((n!)^(1/2) (Pi a^2)^(1/4)))^2*x^2,
          {x, -Infinity, Infinity}, 
          Assumptions :> {Element[n, Integers], Element[a, Reals], a > 0}]

And getting as a result:

  Integrate[(x^2 exp[-(x^2/(2 a^2))]^2 HermiteH[n, x/a]^2)/(Sqrt[a^2] Sqrt[Pi] n!),
            {x, -∞, ∞}, 
            Assumptions :> {n ∈ Integers, a ∈ Reals, a > 0}]

What do I have to change to get an answer?

Answer 1

One can observe that given n Mathematica can evaluate the above integral symbolically assuming e.g. a > 0 (we needn't assume a ∈ Reals since the former assumption implies it). Thus finding e.g. the first 5 integrals we can make use FindSequenceFunction:

intherm[a_, n_] =
  FindSequenceFunction[ 
    Table[{n, Integrate[ E^(-x^2/a^2) x^2 HermiteH[n, x/a]^2/(n! Sqrt[(a^2)] Sqrt[π]), 
                         {x, -∞, ∞}, Assumptions -> a > 0]}, 
          {n, 0, 5}], n] // Simplify

 2^(-1 + n) a^2 (1 + 2 n)

TraditionalForm[2^(-1 + n) a^2 (1 + 2 n)]

Let's check the result for a large number n e.g. n = 131:

With[{n = 131}, 
     Integrate[(E^(-(x^2/a^2)) x^2 HermiteH[n, x/a]^2)/(n! Sqrt[(a^2)] Sqrt[π]), 
               {x, -∞, ∞}, Assumptions -> a > 0] == 2^(-1 + n) a^2 (1 + 2 n)]

True

There are different approaches e.g. exploiting recurrence relations between Hermite polynomials, but this one is quite simple and straightforward therefore there's no need for further considerations of the issue.

Answer 2

It is able to do the integration for fixed n.

hermInt[n_] := Integrate[(HermiteH[n, x/a]*
  Exp[-x^2/(2 a^2)]/((n!)^(1/2) (Pi a^2)^(1/4)))^2* x^2, {x, -Infinity, Infinity}, 
  Assumptions :> {Element[a, Reals], a > 0}]

Here are the first 10:

hermInt[#] & /@ Range[10]
{3 a^2, 10 a^2, 28 a^2, 72 a^2, 176 a^2, 416 a^2, 960 a^2, 2176 a^2, 4864 a^2, 10752 a^2}

If we look at these coefficients they form a nice pattern:

FindSequenceFunction[{3, 10, 28, 72, 176, 416, 960, 2176, 4864, 10752}, n]
2^(-1 + n) (1 + 2 n)