The function H_n(x) represents the n-th Hermite polynomial, which is a type of orthogonal polynomial used in various areas of mathematics and physics. These polynomials are defined using a specific recurrence relation and are particularly important in quantum mechanics, especially in the context of the quantum harmonic oscillator. Hermite polynomials can be expressed in terms of the exponential function and are often used in probability theory and numerical analysis. They are orthogonal with respect to the weight function e^-x^2 over the interval from negative to positive infinity, making them useful in solving differential equations and in approximation theory.