I have solved an eigenvalue problem using Mathematica and the answer is in terms of Hermite polynomials.
Now, for integer degrees $H_n(z)$, I can find a nice definition. However, in the solution to the aforementioned problem, $n$ is not an integer. For example, my Mathematica solution contains HermiteH[a,b] where a and b are both real numbers. The "degree" of the polynomial in this case is a.
Where can I find a definition for this? I suspect I can just generalize the contour integral to use a gamma function instead of a factorial so that
$$
H_a(z) = \frac{\Gamma(a+1)}{2\pi i} \oint_C \frac{e^{2tx-t^2}}{t^{a+1}} \, dt
$$
for a contour $C$ around the origin, vs.
$$
H_n(z) = \frac{n!}{2\pi i} \oint_C \frac{e^{2tx-t^2}}{t^{n+1}} \, dt
$$
where $n$ is a nonnegative integer.
Is this correct, and if so, what's a relatively authoritative source for the generalization?
Ideally, I'd prefer a series representation (or something else a bit more explicit) if it's known.
Best Answer
After some more research, it appears Mathematica is using the following form
$$ H_\nu(z) = 2^\nu\sqrt{\pi}\left( \frac{{}_1F_1\left(-\frac{\nu}{2},\frac{1}{2},z^2 \right)}{\Gamma\left( \frac{1-\nu}{2} \right)} - 2z\frac{{}_1F_1\left( \frac{1-\nu}{2},\frac{3}{2},z^2 \right)}{\Gamma\left( -\frac{\nu}{2} \right)} \right) $$
See https://functions.wolfram.com/HypergeometricFunctions/HermiteHGeneral/02/