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2 Polynomial Approximation of Differential Equations
(1967), dettman(1969). We only remark that, in most of the cases, one can show that
the set of eigenvalues form a divergent sequence of real positive numbers. Many families
of eigenfunctions have been widely studied. Among these, Bessel functions are the most
representative (see section 12.4). However, we are mainly concerned with polynomial
solutions of (1.1.1). In particular, we are interested in introducing and characterizing
sequences {(λn, un)} n∈N of solutions to (1.1.1) with λn > 0, lim
n→∞ λn = + ∞, where
the un’s are polynomials of degree n, suitably normalized. Many examples will be given
in the following. Here we state a first general result, which will be proven in section 2.2.
Theorem 1.1.1 - If {un}n∈N is a sequence of solutions to (1.1.1), where un is a poly-
nomial of degree n, then it is possible to find three real sequences {ρn}, {σn}, {τn}, n ≥ 2
such that
(1.1.2) un (x) = (ρn x + σn) un−1 (x) + τn un−2 (x), ∀n ≥ 2, ∀x ∈ I.
According to (1.1.2), we define ρ1, σ1 ∈ R, such that u1(x) = (ρ1x + σ1)u0(x).
Theorem 1.1.1 allows us to recursively compute the n th polynomial at a given point
x ∈ I, starting from the values u0(x) and u1(x). By differentiating the expression
(1.1.2), we get a similar relation for the derivative, i.e.
(1.1.3) u ′ n (x) = (ρn x + σn) u ′ n−1 (x) + ρn un−1 (x) + τn u ′ n−2 (x), ∀n ≥ 2, ∀x ∈ I.
The simultaneous evaluation of (1.1.2) and (1.1.3) gives u ′ n at the point x. Evaluation
of higher order derivatives can be clearly carried out in the same way.
1.2 The Gamma function
Before continuing our analysis of polynomial eigenfunctions, it is convenient to review
some properties of the well-known Euler Gamma function which, for any real positive