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Special Families of Polynomials 5
By applying the Frobenius method (see for instance dettman(1969)), we get the fol-
lowing result.
Theorem 1.3.1 - The solution to (1.3.1) is a polynomial of degree n only when
λ = n(n + α + β + 1), n ∈ N.
We shall define the n-degree Jacobi polynomial P (α,β)
n
(1.3.1) normalized by
(1.3.2) P (α,β)
n (1) :=
or equivalently by
n + α
=
n
Γ(n + α + 1)
(1.3.3) P (α,β)
n (−1) := (−1) n
n! Γ(α + 1)
to be the unique solution of
, n ∈ N,
n + β
, n ∈ N.
n
We observe that no boundary conditions are imposed in (1.3.1). These are replaced
by requiring the solutions to be polynomials. Condition (1.3.2) (or (1.3.3)) is only
imposed to select a unique eigenfunction, otherwise defined to within a multiplicative
constant.
Many theorems and properties for this family of polynomials are well-known. Here
we just collect some basic results and we refer to szegö(1939), luke(1969), askey(1975),
srivastava and manocha(1984) for a detailed essay. First we give the Rodrigues’ for-
mula:
(1.3.4) (1 − x) α (1 + x) β P (α,β)
n (x) = (−1)n
2n n!
A more explicit form is
(1.3.5) P (α,β)
n (x) = 2 −n
=
Γ(2n + α + β + 1)
2 n n! Γ(n + α + β + 1)
n
n + α n + β
k n − k
k=0
x n +
dn dxn n+α n+β
(1 − x) (1 + x) , n ∈ N.
(x − 1) n−k (x + 1) k
(α − β) n
2n + α + β xn−1
+ · · · , n ∈ N.