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4 Polynomial Approximation of Differential Equations
(1.2.5) Γ(x) =
where
τ0 :=
Hk(x) :=
x + 9
1 x− 2
2
2
11 , τk := 2(−1) k k
e 1−x
τ0 +
∞
k=1
(x − 1) (x − 2) · · · (x − k)
x(x + 1) · · · (x + k − 1)
k
m=0
(k + m − 1)!
m!(k − m)!
τk Hk (x)
, k ≥ 1 ,
,
(−1) m em (m + 11 , k ≥ 1.
2
)m+1/2
This is basically the expression given in luke(1969), Vol.1, page 30. More theoretical
details and tabulated values are contained in that book.
1.3 Jacobi polynomials
We introduce the first family of polynomial solutions to (1.1.1) called Jacobi polynomi-
als. They depend on two parameters α, β ∈ R with α > −1, β > −1. As we will see
in the following, an appropriate choice of these parameters leads to other well-known
families, such as Legendre or Chebyshev polynomials.
Let us set I =] − 1,1[ and let us take a, b, w in (1.1.1) such that
a(x) = (1 − x) α+1 (1 + x) β+1 , ∀x ∈ Ī,
b(x) = 0 , w(x) = (1 − x) α (1 + x) β , ∀x ∈ I,
α > −1, β > −1.
After simplification, this choice yields the singular eigenvalue equation
(1.3.1) − 1 − x 2 u ′′ + (α + β + 2)x + α − β u ′ = λu.