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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.8 Differential Equations

See Table 18.8.1 and also Table 22.6 of Abramowitz and Stegun (1964).

Table 18.8.1: Classical OP’s: differential equations A(x)f′′(x)+B(x)f(x)+C(x)f(x)+λnf(x)=0.
f(x) A(x) B(x) C(x) λn
Pn(α,β)(x) 1-x2 β-α-(α+β+2)x 0 n(n+α+β+1)
(sin12x)α+12(cos12x)β+12×Pn(α,β)(cosx) 1 0 14-α24sin212x+14-β24cos212x (n+12(α+β+1))2
(sinx)α+12Pn(α,α)(cosx) 1 0 (14-α2)/sin2x (n+α+12)2
Cn(λ)(x) 1-x2 -(2λ+1)x 0 n(n+2λ)
Tn(x) 1-x2 -x 0 n2
Un(x) 1-x2 -3x 0 n(n+2)
Pn(x) 1-x2 -2x 0 n(n+1)
Ln(α)(x) x α+1-x 0 n
e-12x2xα+12Ln(α)(x2) 1 0 -x2+(14-α2)x-2 4n+2α+2
Hn(x) 1 -2x 0 2n
e-12x2Hn(x) 1 0 -x2 2n+1
Hen(x) 1 -x 0 n