Family
Legendre P (w=1 on [-1,1])
Chebyshev T (w=1/√(1-x²))
Hermite H (w=e^(-x²), physicists)
Laguerre L (w=e^(-x) on [0,∞))
Degree m
Degree n
⟨f
m
,f
n
⟩ (numeric) =
Analytic =
Independent check
Equations
Recurrences: Legendre (k+1)P
k+1
=(2k+1)xP
k
−kP
k−1
; Chebyshev T
k+1
=2xT
k
−T
k−1
; Hermite H
k+1
=2xH
k
−2kH
k−1
; Laguerre (k+1)L
k+1
=(2k+1−x)L
k
−kL
k−1
. Norms: 2/(2n+1); π/2 (n>0), π (n=0); 2
n
n!√π; 1.