- Authors: D. van Melkebeek and A. Bultheel.
- Reference: Journal of Computational and Applied Mathematics,
49(1-3): 305-316, 1993.
To study polynomials orthogonal with respect to the logarithmic
equilibrium measure on the Julia set of a nonlinear polynomial P,
Daniel Bessis and his coworkers introduced a symmetric bilinear form,
the symmetric P-form. It is a nonhermitian variant of a genuine
inner product, the hermitian P-form. The main reason for doing so,
was the existence of a three term recurrence relation between the
polynomials of successive degree of an orthogonal system.
However, this is only true if the moment matrix of the symmetric
P-form is strongly regular. In this paper, we investigate what
happens if this condition is not met. We distinguish between the case
where an orthogonal system still exists and the one where it does not,
and illustrate them by
P(z) = z2 and P(z) = z2 - 1