Matrices and orthogonal polynomials

The Lanczos algorithm of minimized iterations shows that a polynomial verifying a three-term recurrence relation can be written as the determinant of a tridiagonal matrix, here we exhibit examples of this property. Besides, for several orthogonal polynomials, Cohen proved that their roots are the proper values of symmetric tridiagonal matrices; here we give examples of this Cohen’s result for the Legendre, Laguerre, and Hermite polynomials, which are important in applications to numerical analysis and quantum mechanics.