Derivation From Differential Equation
All of the polynomial sequences arising from the differential equation above are equivalent, under scaling and/or shifting of the domain, and standardizing of the polynomials, to more restricted classes. Those restricted classes are exactly "classical orthogonal polynomials".
- Every Jacobi-like polynomial sequence can have its domain shifted and/or scaled so that its interval of orthogonality is, and has Q = 1 − x2. They can then be standardized into the Jacobi polynomials . There are several important subclasses of these: Gegenbauer, Legendre, and two types of Chebyshev.
- Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and/or reflected so that its interval of orthogonality is, and has Q = x. They can then be standardized into the Associated Laguerre polynomials . The plain Laguerre polynomials are a subclass of these.
- Every Hermite-like polynomial sequence can have its domain shifted and/or scaled so that its interval of orthogonality is, and has Q = 1 and L(0) = 0. They can then be standardized into the Hermite polynomials .
Because all polynomial sequences arising from a differential equation in the manner described above are trivially equivalent to the classical polynomials, the actual classical polynomials are always used.
Read more about this topic: Classical Orthogonal Polynomials
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