Hermite Polynomials are a set of orthogonal polynomials that arise in probability, combinatorics, and physics. They are defined using a specific recurrence relation and are often denoted as H_n(x), where n is a non-negative integer. These polynomials are particularly important in quantum mechanics, especially in the context of the quantum harmonic oscillator. The properties of Hermite Polynomials include their orthogonality with respect to the weight function e^{-x^2} over the interval from negative to positive infinity. They also satisfy a differential equation known as the Hermite differential equation, making them useful in various applications, including numerical analysis and solving differential equations.