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Is there a solution to this definite integral

$$ \int_{-\infty}^{d*} \left(\sqrt{c_0\; x^2 +c_1}\right)^{-m} H_{-m}\left(c_2\; \sqrt{\frac{c_0\; x^2}{c_0\; x^2 + c_1}}\right) \mathrm{d} x $$

with $m$ a positive integer and $c_0$, $c_1$, and $c_2$ constants where $H$ is the Hermite polynomial?

I am using the physics version of the Hermite polynomial.

Gary
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Denis Cousineau
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  • What is a Hermite polynomial with negative order? – Gary May 06 '22 at 13:22
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    $d$ is probably one of the worst choices of letter for a variable of integration. – eyeballfrog May 06 '22 at 13:47
  • I am sorry that the variable examined is known as $d$ in my research community (cognitive neurosciences). I'll send a request to rename it. – Denis Cousineau May 06 '22 at 15:39
  • @Gary : One definition of Hermite polynomials are those that satisfy the differential equation $y''-2x y'+2n y=0$. Although the Wiki page fails to mention it, nothing prevent $n$ from being negative. See this SE response. – Denis Cousineau May 06 '22 at 15:45
  • @DenisCousineau Yes, but then they are no longer polynomials. I changed your integration variable from $d$ to $x$. I hope you don't mind. Since we don't use the context here, I belive it does not matter. – Gary May 06 '22 at 22:24

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