# Polynomials satisfying a three-term recurrence

Let ${p_n}(x) = x{p_{n - 1}}(x) - {a_{n - 2}}{p_{n - 2}}(x)$ for some numbers ${a_n}$ with initial values ${p_{ - 1}}(x) = 0$ and ${p_0}(x) = 1.$ By Favard’s theorem about orthogonal polynomials there exists a linear functional $F$ on the vector space of polynomials such that
$$F\left( {{p_n}(x){x^k}} \right) = 0$$ for $0 \le k < \deg {p_n}$

and

$$\det \left( {F\left( {{x^{i + j}}} \right)} \right)_{i,j = 0}^{\deg {p_n} - 1} \ne 0.$$

It seems that analogous results also hold if ${p_n}(x) = {x^{{r_{n - 1}}}}{p_{n - 1}}(x) - {a_{n - 2}}{p_{n - 2}}(x),$ where ${r_0} = 1$ and ${r_n}$ are arbitrary positive integers for $n > 0.$
I do not know where to look for such results. Therefore I would be very grateful for references.

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Is it easy to write down $F$ for the Cheybyshev polynomials, when $a_{n-2}=1$? – Peter Samuelson Mar 4 '13 at 17:18
@ Peter: For the Chebyshev polynomials of the first kind $F$ is given as an integral on the interval [-1,1] with weight $\frac{1}{{\sqrt {1 - {x^2}} }}$, for the Chebyshev polynomials of the second kind with weight $\sqrt {1 - {x^2}} .$ – Johann Cigler Mar 4 '13 at 17:56