I recently asked a question about a differential equation, and received this as an answer. It included a Hermite polynomial of negative degree, namely $H_{-3}$. I searched online and it seems as though these $H_n$'s are only defined for $n\ge0$, and for $n<0$ something else might used – parabolic cylinder functions. I say 'might' because I am not sure if they are actually the same. I have never heard of these, and so was hoping someone would know about them here. Can they be expressed in terms of $erf$ or $erfc$? I found another link which suggested they could. If so, how?
To clarify I want to know if Hermite polynomials (or their equivalent) can be expressed for negative integers $n$ in some closed form expression (including $erf$).
Best Answer
As I added to my answer to the previous question.
Expanded using more usual functions, $$H_{-3}\left(\frac{x}{2}\right)=\frac{1}{16} \left(\sqrt{\pi } e^{\frac{x^2}{4}} \left(x^2+2\right) \text{erfc}\left(\frac{x}{2}\right)-2 x\right)$$
Similarly,$$H_{-1}\left({x}\right)=\frac{1}{2} \sqrt{\pi } e^{x^2} \text{erfc}(x)$$ $$H_{-2}\left({x}\right)=\frac{1}{2} \left(1-\sqrt{\pi } e^{x^2} x \text{erfc}(x)\right)$$ $$H_{-3}\left({x}\right)=\frac{1}{8} \left(\sqrt{\pi } e^{x^2} \left(2 x^2+1\right) \text{erfc}(x)-2 x\right)$$ $$H_{-4}\left({x}\right)=\frac{1}{24} \left(2 \left(x^2+1\right)-\sqrt{\pi } e^{x^2} x \left(2 x^2+3\right) \text{erfc}(x)\right)$$ $$H_{-5}\left({x}\right)=\frac{1}{192} \left(\sqrt{\pi } e^{x^2} \left(4 \left(x^2+3\right) x^2+3\right) \text{erfc}(x)-2 x \left(2 x^2+5\right)\right)$$