Three-term recurrences

There many examples of families of functions where each function can be computed as a linear combination of the two previous terms where a and b are functions of x but not on n. This is called a three-term recurrence formula. It’s amazing how often you can run into three-term recurrence formulas. There are theorems that give conditions for such recurrences to hold, but I haven’t reached the bottom of that rabbit hole [1]. For this post I just want to give examples. Bessel functions of the first and second kind: Modified Bessel functions of the first and second kind: Chebyshev polynomials of the first and second kind: Hermite polynomials (physicists’ convention): Legendre polynomials: [1] See Bochner’s theorem for orthogonal polynomials, the Nikiforov–Uvarov method, and Infeld-Hull factorization.The post Three-term recurrences first appeared on John D. Cook.