Ratios of orthogonal polynomials

[Alan Horwitz <alh4@psu.edu>]

Let p(x) be a polynomial with all real zeroes 
r_1 < r_2 < ... < r_n and critical points 
x_1 < x_2 < ... < x_(n-1). Define the ratios
s_k = (x_k-r_k)/(r_(k+1)-r_k), k = 1,2,...,n-1. I have recently done some
research in this area. One of the questions I was interested in was the
monotonicity fo the ratios. I proved that for n = 4(n = 3 is trivial) the
ratios are monotonic-i.e. s_1 < s_2 < s_3, while for n>=5, the ratios are
not monotonic in general. I now want to investigate properties of the
ratios of some of the classical orthogonal polynomials. In particular, I
can prove that for the Chebyshev polynomials T_n:Let s_k,n denote the kth
ratio of T_n. Then s_k,n < s_k+1,n(so the ratios are increasing for fixed
n) and s_k,n > s_k,n+1(so the ratios are decreasing functions of n). For
some of the other classical orthogonal polynomials this is not as easy to
show(numerical evidence indicates it's true for the Legendre polynomials)
since one does not have explicit formulas for the roots and critical
points. Has anyone seen results of this type before ? Is this sort of
result interesting to those doing research in orthogonal polynomials ? I'm
looking for good upper and lower bounds on r_k and x_k(as functions of n
and k) which might enable me to prove more general results. 

Dr.Alan Horwitz
Penn State University
25 Yearsley Mill Rd.
Media, PA 19063
(610)-892-1449
alh4@psu.edu
Home Page: http://www.math.psu.edu/horwitz/

"A Mathematician is a machine for turning coffee into theorems."-Erdos