Secondary polynomials - Alchetron, The Free Social Encyclopedia

In mathematics, the secondary polynomials { q n ( x ) } associated with a sequence { p n ( x ) } of polynomials orthogonal with respect to a density ρ ( x ) are defined by

q n ( x ) = ∫ R p n ( t ) − p n ( x ) t − x ρ ( t ) d t .

To see that the functions q n ( x ) are indeed polynomials, consider the simple example of p 0 ( x ) = x 3 . Then,

q 0 ( x ) = ∫ R t 3 − x 3 t − x ρ ( t ) d t = ∫ R ( t − x ) ( t 2 + t x + x 2 ) t − x ρ ( t ) d t = ∫ R ( t 2 + t x + x 2 ) ρ ( t ) d t = ∫ R t 2 ρ ( t ) d t + x ∫ R t ρ ( t ) d t + x 2 ∫ R ρ ( t ) d t

which is a polynomial x provided that the three integrals in t (the moments of the density ρ ) are convergent.

References

Secondary polynomials Wikipedia

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