This post provides a couple examples to go along with two earlier posts.

The pattern we’re illustrating is families of polynomials p_n(x) that each satisfy a differential equation and a three-term recurrence. The differential equations have a second solution q_n(x) that is the larger solution with respect to x but the smaller solution with respect to n.

In both the examples below p_n(x) is a polynomial, and so bounded on the interval [−1, 1], and q_n(x) is not a polynomial, with singularities at ±1. This is analogous to the previous examples with Bessel functions J_n(x) and Q_n(x) that satisfy the same differential equation but have contrasting behavior with respect to x versus n.

Legendre polynomials

The differential equation

has two solutions for each n, P_n(x) and Q_n(x).

The solutions P_n(x) are the Legendre polynomials. The solutions Q_n(x) are not polynomials but involve a term log((1 + x)/(1 − x)) that blows up at 1 and −1. But for fixed x and increasing n, P_n(x) grows exponentially and Q_n(x) decays exponentially.

Chebyshev polynomials

The differential equation

has two solutions for each n, T_n(x) and V_n(x).

The solutions T_n(x) are the Chebyshev polynomials. The solutions V_n(x) are not polynomials but involve a term √(1 — x²) that become vertical up at 1 and −1. But for fixed x and increasing n, T_n(x) grows exponentially and V_n(x) decays exponentially.