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Orthogonality 29
Therefore, one obtains
(2.3.9) bk =
⎧
⎪⎨
⎪⎩
cn
ρn+1
cn−1
ρn
ck−1
ρk
− c0σ1
ρ1
− cnσn+1
ρn+1
− ckσk+1
ρk+1
− c1τ2
ρ2
− ck+1τk+2
ρk+2
k = n + 1,
k = n,
1 ≤ k ≤ n − 1,
k = 0.
For the various families of polynomials, the sequences {ρn}, {σn}, {τn} are respectively
given by (1.3.8), (1.4.2), (1.5.2), (1.6.5) and (1.7.6) (ρ1 and σ1 are defined according to
u0 and u1). For ultraspherical and Hermite polynomials we have σn = 0, ∀n ≥ 1.
For more general functions g the answer can be very complicated and more explicit
expressions are often unavailable. An interesting case is g(x,p(x)) = p 2 (x). For ultra-
spherical polynomials the following relation holds
(2.3.10) P (α,α)
k P (α,α)
j =
ν Γ(k + ν + 1
2
min(k,j) k + j + ν − 2m
×
(m + ν)(k − m + ν)(j − m + ν)
m=0
k + j − 2m + 2ν
×
k + j − 2m
where ν = α + 1
2 .
−1
1 ) Γ(j + ν + 2 ) Γ(ν + 1)
2 Γ(k + 2ν) Γ(j + 2ν) Γ(ν + 1
2 )
m + ν k − m + ν j − m + ν
m k − m j − m
Γ(k + j − m + 2ν) Γ(k + j − 2m + 2ν + 1)
Γ(k + j − m + ν + 1)
k+j−2m
Γ(k + j − 2m + ν + 1
(α,α)
P
2 )
α > −1, α = −1 2 , k,j ∈ N,
,