# A special solution to the Hermite Differential Equation

I know that the general form solution to the Hermite differential equation $$y''-2xy'+2\lambda y=0$$ is $$y(x)=a_1 M(-\frac{\lambda}{2},\frac{1}{2},x^2)+a_2 H(\lambda,x),$$ where $$M(\cdot,\cdot,\cdot)$$ is a confluent hypergeometric function of the first kind, and $$H(\cdot,\cdot)$$ is a Hermite polynomial.

For a general value of $$\lambda$$ (negative and non-integer real valued), is there a special solution to the Hermite differential equation such that it's first order derivative goes to zero for $$x\rightarrow-\infty$$? In other words, is there a parametric characterization of $$a_2$$ as a function of $$a_1$$ and $$\lambda$$ such that $$y'(x)\rightarrow 0$$ as $$x\rightarrow-\infty$$?

From mathematica numerical calculations it seems that such a special case exists. However, I was not able to characterize it explicitly. I would appreciate any help on this. Thank you!

$$y(x)=a_1 M(-\frac{\lambda}{2},\frac{1}{2},x^2)+a_2 H(\lambda,x)\Rightarrow$$ $$y'(x)=-2a_1\lambda x M(1-\tfrac{1}{2}{\lambda},\tfrac{3}{2},x^2)+2 {a_2} {\lambda} H({\lambda}-1,x)$$ The asymptotics for $$x\rightarrow-\infty$$ and $$\lambda$$ negative non-integer is $$y'(x)\rightarrow\frac{2 e^{x^2}(-x)^{-\lambda} }{\sqrt{\pi }\, \Gamma \left(-\lambda/2\right)}\bigl(a_2 \Gamma \left(-\lambda/2\right) \Gamma (\lambda+1)\sin \pi \lambda -\pi a_1\bigr).$$ So this vanishes if $$a_2 \Gamma \left(-\lambda/2\right) \Gamma (\lambda+1)\sin \pi \lambda =\pi a_1.$$
• Thank you Carlo. Why does the $a_1$ term diverge as $e^{\frac{x^2}{2}}$? From en.wikipedia.org/wiki/… it seems that it should diverge as $e^{x^2}$? Also why does the $a_2$ term diverge as $e^{x^2}$? I couldn't find any pointers for this. I really appreciate your help!! Commented Mar 25, 2019 at 18:19
• Thank you. Can you kindly point to me the reference of the asymptotics for H(,)? I couldn't find anything for negative non-integer $\lambda$. Would really appreciate your help. Commented Apr 3, 2019 at 20:12