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I would like to understand $L^2(\mathbb{R},\mu)$ approximation by polynomials of $\tanh$ up to degree $n$ where $\mu$ is the standard Gaussian distribution. This leads to considering the Hermite expansion of $\tanh$ and looking at the rate of decay of the coefficients. Is the tail behavior of the Hermite expansion of $\tanh$ understood? That is, if we truncate the Hermite expansion at degree $n$ then what is the error?
In other words, consider the integral $$ a_n = \int_{-\infty}^{\infty} H_n(x) e^{-x^2/2} \tanh(x) d\mu $$ where $H_n$ is the $n$th normalized Hermite polynomial. What is the asymptotics of $a_n$?

One approach is express the derivatives of $\tanh$ as polynomials of $\tanh$. This is possible since $\frac{d}{dx} \tanh(x) = 1 -\tanh^2(x)$. Is this sequence of polynomials correspond to some known sequence of polynomials? For example, can we bound the maximum modulus of the polynomial family over $[-1,1]$.

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$$|a_n|\approx\frac{\Gamma(n/2+1)}{\Gamma(n+1)}\exp\left(-\frac{\pi\sqrt{2n}}{2}\right).$$ Ref. Szego, Orthogonal polynomials, AMS. 1959, formula (8.23.4), see also (9.2.9).

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