Secondary polynomials
In mathematics, the secondary polynomials { q n ( x ) } {\displaystyle \{q_{n}(x)\}} associated with a sequence { p n ( x ) } {\displaystyle \{p_{n}(x)\}} of polynomials orthogonal with respect to a density ρ ( x ) {\displaystyle \rho (x)} are defined by q n ( x ) = ∫ R p n ( t ) − p n ( x ) t − x ρ ( t ) d t . {\displaystyle q_{n}(x)=\int _{\mathbb {R} }\! {\frac {p_{n}(t)-p_{n}(x)}{t-x}}\rho (t)\,dt.} To see that the functions q n ( x ) {\displaystyle q_{n}(x)} are indeed polynomials, consider the simple example of p 0 ( x ) = x 3 .