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In mathematics, the secondary polynomials \ associated with a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
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polynomials In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
with respect to a density \rho(x) are defined by : q_n(x) = \int_\mathbb\! \frac \rho(t)\,dt. To see that the functions q_n(x) are indeed polynomials, consider the simple example of p_0(x)=x^3. Then, :\begin q_0(x) & = \int_\mathbb \! \frac \rho(t)\,dt \\ & = \int_\mathbb \! \frac \rho(t)\,dt \\ & = \int_\mathbb \! (t^2+tx+x^2)\rho(t)\,dt \\ & = \int_\mathbb \! t^2\rho(t)\,dt + x\int_\mathbb \! t\rho(t)\,dt + x^2\int_\mathbb \! \rho(t)\,dt \end which is a polynomial x provided that the three integrals in t (the moments of the density \rho) are convergent.


See also

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Secondary measure In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal ...
Polynomials {{algebra-stub