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In the study of a statistical physics problem, I need to know the orthogonal polynomials with respect to the weight $$2\cosh(\beta x)e^{-x^2},$$ where $\beta \in \mathbb{R}^+$. Is this already known?

There is also a related but probably harder problem: what are the the orthogonal polynomials with respect to the weight $$\ln [2\cosh(\beta x)]e^{-x^2},$$ where $\beta \in \mathbb{R}^+$?

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    $\begingroup$ What are the first few that you've gotten by Graham-Schmidt from the $x^n$? $\endgroup$
    – AHusain
    Commented Jan 31, 2018 at 22:49
  • $\begingroup$ I assumed that the integration is from $-\infty$ to $\infty$. To simplify notation I changed the weight function to $\frac{2}{\sqrt{\pi}}\cosh(\sqrt{2 b} x) e^{-x^2-b/2}$ and got by Gram-Schmidt the first polynomials as: $2^{-1/2}, \frac{x}{\sqrt{b+1}}, \frac{2 x^2-b-1}{2\sqrt{1+2 b}}, ...$. The higher degree polynomials are to bulky to be listed in a comment. $\endgroup$
    – Johannes Trost
    Commented Feb 1, 2018 at 16:35

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