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In mathematics, the secondary polynomials { q n } {\displaystyle \{q_{n}\}} associated with a sequence { p n } {\displaystyle \{p_{n}\}} of polynomials orthogonal with respect to a density ρ {\displaystyle \rho } are defined by
To see that the functions q n {\displaystyle q_{n}} are indeed polynomials, consider the simple example of p 0 = x 3 . {\displaystyle p_{0}=x^{3}.} Then,
which is a polynomial x {\displaystyle x} provided that the three integrals in t {\displaystyle t} are convergent.