The n-th Hermite polynomial is a specific type of orthogonal polynomial that arises in probability, combinatorics, and physics. It is denoted as H_n(x) and is defined using a recurrence relation or through its explicit formula involving the exponential function. These polynomials are particularly important in the context of quantum mechanics, where they describe the wave functions of the quantum harmonic oscillator. Hermite polynomials possess several key properties, including 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, which makes them useful in various applications, including statistical mechanics and signal processing.