Differential equation Hermite polynomials are a type of special function that arise in solving certain types of differential equations. They are named after French mathematician Charles Hermite and are denoted by the symbol Hn(x), where n is a non-negative integer.
Hermite polynomials have a wide range of applications in physics and engineering, particularly in the areas of quantum mechanics, statistical mechanics, and signal processing. They are also used in solving problems related to heat conduction, quantum harmonic oscillators, and eigenvalue problems.
Hermite polynomials are defined recursively using the following formula: H0(x) = 1, H1(x) = 2x, and Hn+1(x) = 2xHn(x) - 2nHn-1(x) for n ≥ 1. They can also be expressed using the generating function: ∑n=0∞ Hn(x)zn = e2xz - z2.
Some of the key properties of Hermite polynomials include orthogonality, recurrence relation, generating function, and Rodrigues' formula. They are also symmetric and have only real roots. Additionally, they can be expressed in terms of other special functions such as gamma functions and Bessel functions.
Aside from their use in solving differential equations, Hermite polynomials have numerous applications in physics. They are used to describe the wave functions of quantum systems, such as the ground state of a particle in a harmonic oscillator potential. They also play a role in the description of the energy levels of the hydrogen atom and in calculating the partition function in statistical mechanics.