Classical Sturmian sequences

Abstract

The Sturm sequence is generated by a pair of polynomials $P(x)$ and $P'(x)$, where $P(x)$ is assumed to have simple real roots. Euclidean algorithm generates then a finite sequence of polynomials orthogonal on the grid $x_s$ of roots of the polynomial $P(x)$. This algorithm can be exploited in order to find the number of roots of the polynomial $P(x)$ inside a given interval. We consider the "inverse" problem: what is the explicit system of orthogonal polynomials generated by the prescribed grid $x_s$ of "classical" type. The main results are the following. The generic linear grid generates a special case of the Hahn polynomials. The quadratic grids $x_s=x(s+1)$ and $x_s=s(s+2)$ correspond to two special cases of the Racah polynomials. The generic exponential grid is related to a special case of the q-Hahn polynomials. Finally, we show that two special trigonometric grids are related to the Chebyshev polynomials of the first and second kind.