If p(z) is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of p′(z) lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of p(z) to a nearest zero of p′(z)? We obtain bounds for this distance depending on degree. We also show t...

Exceptional orthogonal polynomials were introduced by Gomez-Ullate, Kamran and Milson as polynomial eigenfunctions of second order differential equations with the remarkable property that some degrees are missing, i.e., there is not a polynomial for every degree. However, they do constitute a complete orthogonal system with respect to a weight function that is typically a rational modification ...

Sharp bounds are given for the highest multiplicity of zeros polynomials in terms their norm on Jordan curves and arcs. The results extend a theorem Erdős Turán solve problem them from 1940.

There is an interesting analogy between the description of the real square roots of 3×3 matrices and the zeros of the (depressed) real quartic polynomials. This analogy, which in fact better explains the nature of the zeros of those polynomials, is unveiled through a natural use of the Cayley-Hamilton theorem.

We use a method based on the division algorithm to determine all the values of the real parameters b and c for which the hypergeometric polynomials 2F1(−n, b; c; z) have n real, simple zeros. Furthermore, we use the quasi-orthogonality of Jacobi polynomials to determine the intervals on the real line where the zeros are located.

We investigate the interlacing of the zeros of linear combinations pn+ aqm with the zeros of the components pn and qm, where {pn} ∞ n=0 and {qm} ∞ m=0 are different sequences of Jacobi polynomials. The results we prove hold when pn and qm are Jacobi polynomials P (α,β) n (x) and P (α,β) m (x) for certain values of α and β with m = n or m = n−1. Numerical counterexamples are given in situations ...

We study the limiting behavior of the zeros of the Euler polynomials. When linearly scaled, they approach a definite curve in the complex plane related to the Szegö curve which governs the behavior of the roots of the Taylor polynomials associated to the exponential function. Further, under a conformal transformation, the scaled zeros are uniformly distributed.