Hermite Differential Equation - Non-integer values of $\lambda$

Question

The Hermite differential equation, given by : $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ has solutions of the $$ y(x) = \mathcal{H_n(x)} $$ when $ \lambda \: \epsilon \:\mathcal{Z_+} $

Are there solutions to this equation for a more general case, where $ \lambda $ is a real number ? More importantly are there convergent solutions for cases where $ \lambda $ is not integer.

Answer 1

I suggest you have a look at
http://mathworld.wolfram.com/HermiteDifferentialEquation.html

The general solution of this equation is
$$c_1 H_{\frac{\lambda }{2}}(x)+c_2 \, _1F_1\left(-\frac{\lambda }{4};\frac{1}{2};x^2\right)$$