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6 Polynomial Approximation of Differential Equations
Both (1.3.4) and (1.3.5) can be checked by direct substitution in (1.3.1). A lot of
formulas relate Jacobi polynomials corresponding to different choices of the pair of
parameters (α, β). We show one of the most representative, i.e.
(1.3.6)
d
dx
P (α,β)
n
= 1
2
By (1.3.4) or (1.3.5) it is clear that
(α, β)
(1.3.7) P
0 (x) = 1 , P
(n + α + β + 1) P (α+1, β+1)
n−1 , n ≥ 1.
(α, β)
1
(x) = 1
1
(α + β + 2)x + (α − β).
2 2
Since (1.3.4) and (1.3.5) are quite unpracticable, higher degree polynomials can be
determined using theorem 1.1.1. More precisely, we have
(1.3.8) ρn =
τn = −
σn =
(2n + α + β) (2n + α + β − 1)
,
2n(n + α + β)
(α 2 − β 2 ) (2n + α + β − 1)
2n (n + α + β) (2n + α + β − 2) ,
Moreover, one has: ρ1 = 1
2 (α + β + 2), σ1 = 1
2
using (1.1.3) or (1.3.6).
(n + α − 1) (n + β − 1) (2n + α + β)
, n ≥ 2.
n (n + α + β) (2n + α + β − 2)
(α − β). Derivatives can be recovered
Finally, we present the following estimate in the interval Ī (see szegö (1939)):
(1.3.9) max
−1≤x≤1
|P (α,β)
n
(x) | = max |P (α,β)
n (±1)| = max
n + α
n
,
n + β
n
,
n ∈ N.
Further properties can be proven for the so called ultraspherical (or Gegenbauer)
polynomials. These are Jacobi polynomials where α = β. We soon see two celebrated
examples.