This talk will compare two methods for generating a discrete set of Hermite functions. The continuous-time Hermite functions are well known as the solution of a particular differential equation, but they are also eigenfunctions of the Fourier transform as well as an orthogonal set of functions that can be useful in signal representations. For the discrete case, the methods compared in this generate the discrete Hermite functions as eigenvectors of a matrix that commutes with a Fourier matrix. One method involves a strictly tridiagonal commuting matrix while the other involves a nearly tridiagonal form. Each one can be related to a difference equation. A "centered" Fourier matrix has certain advantages to the generation of the eigenvectors. The discrete fractional Fourier transform (FRFT) is also a focus of this talk, as the FRFT may be computed using discrete Hermite functions. Background concerning the continuous time FRFT will be presented as well as applications of the discrete case.