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After solving a differential equation I've gotten a solution given as a linear combination of Hermite polynomials and confluent hypergeometric functions. The caveat is that the Hermite polynomials that I am encountering are somehow a generalized version, where the degree of the polynomial is negative and fractional, i.e., $H_{-\frac{1}{2}}(x)$. The solution to the differential equation was given by Mathematica, and using the "Wolfram functions site" documentation I was also able to prove it by hand, so it seems that those generalized Hermite polynomials are well defined. For computational purposes I want to use the following formula: (https://functions.wolfram.com/Polynomials/HermiteH/26/01/02/0001/)

$$ H_{\nu}(z)=2^\nu \sqrt\pi\left(\frac{1}{\Gamma(\frac{1-\nu}{2})}{}_{1}F_{1}(\frac{-\nu}{2},\frac{1}{2},z^2) -\frac{2z}{\Gamma(\frac{-\nu}{2})}{}_{1}F_{1}(\frac{1-\nu}{2},\frac{3}{2},z^2) \right) $$

However I don't know whether or not I can just trust the Wolfram documentation. I haven't found the formula in any other site. The generalization of the polynomials is so far obscure to me. Does someone know how this generalization is done?

Thanks!

Hey
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    The rhs is an entire function of $z$ (for integral values of $\nu$ as well, with the convention that one of the terms disappears when $\nu \in \mathbb N^0$). So this is the unique choice if we define $H_\nu(z)$ as an analytic solution of an ODE or as the analytic function coinciding with $2^\nu U(-\nu/2, 1/2, z^2)$ for $z > 0$. – Maxim Nov 11 '22 at 00:11

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