Parabolic cylinder function

The parabolic cylinder functions, often denoted as \( U_n(x) \) and \( V_n(x) \), are special functions that arise in various applications, particularly in mathematical physics and solutions to certain differential equations. They are solutions to the parabolic cylinder differential equation, which is given by: \[ \frac{d^2 y}{dx^2} - \frac{1}{4} x^2 y = 0.