Athabasca University
Athabasca, Alberta, Canada
Empirical studies of a roots of a family of polynomials satisfying the recurrence relation $p(n+1,x) = xp(n,x) - p(m-1,x)$ were carried out for $n<25$. To within a scale factor these are Chebysc polynomials of the second kind. Much is known about these polynomials, but studies of the distribution of roots do not seem to have been made. Several theorems are proved, and some interesting conjectures are made. Geometrically, roots of $p(n,x)$ are found to appear as ratios defined within regular $n$-gons and star $n$-gons.