Hi guys,

I have recently started looking at polynomials $q_n$ generated by initial choices $q_0=1$, $q_1=x$ with, for $n\geq 0$, some recurrence formula

$$q_{n+2}=xq_{n+1}+c_n q_n$$

where $c_n$ is some function in $n$. The first few of these are

$$q_2=x^2+c_0$$ $$q_3=x^3+(c_0+c_1)x$$ $$q_4=x^4+(c_0+c_1+c_2)x^2+c_2c_0$$ $$q_5=x^5+(c_0+c_1+c_2+c_3)x^3+(c_0c_2+c_0c_3+c_1c_3)x$$ $$q_6=x^6+(c_0+c_1+c_2+c_3+c_4)x^4+(c_0c_2+c_0c_3+c_0c_4+c_1c_3+c_1c_4+c_2c_4)x^2$$$$+c_0c_2c_4$$

My question is whether there is a name for the coefficients of the powers of $x$. I realise that they can be written as certain formulations of elementary symmetric polynomials but I am ideally looking for a reference where the specific expressions are studied

Any help would be great :)

orthogonal polynomials, that are related -orthogonal polynomials satisfy a three terms linear recurrence of a slightly more general form, and well characterized. Are you interested in the results with $any$ sequence $(c_n)$ or is your $(c_n)$ a particular sequence? (in which case it should be healthy to check if it corresponds to an orthogonal sequence). – Pietro Majer Dec 30 '10 at 17:23