Hermite polynomials are a sequence of orthogonal polynomials that arise in probability, combinatorics, and physics. They are defined using a specific recurrence relation and can be expressed in terms of the exponential function and Gaussian functions. These polynomials are particularly useful in solving problems related to the quantum harmonic oscillator in quantum mechanics. The n-th Hermite polynomial, denoted as H_n(x), has important properties, including orthogonality with respect to the weight function e^(-x²) over the entire real line. They also play a significant role in numerical analysis and approximation theory, where they are used to construct Gaussian quadrature rules for numerical integration.