# Hermite Transform of Tanh

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 $$[-1,1]$$.

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