Polynomials whose coefficients are successive derivatives of a class of Jacobi polynomials evaluated at x = 1 are stable. This yields a novel and short proof of the known result that the Bessel polynomials are stable polynomials. Stability preserving linear operators are discussed. The paper concludes with three open problems involving the distribution of zeros of polynomials.

It is shown that ∑m j=−m(−1) f(x−j)(f(x+j) (m−j)!(m+j)! ≥ 0, m = 0, 1, ..., where f(x) is either a real polynomial with only real zeros or an allied entire function of a special type, provided the distance between two consecutive zeros of f(x) is at least √ 4− 6 m+2 . These inequalities are a surprisingly similar discrete analogue of higher degree generalizations of the Laguerre and Turan inequ...

We study the expected number of zeros $$P_n(z)=\sum_{k=0}^n\eta_kp_k(z),$$ where $\{\eta_k\}$ are complex-valued i.i.d standard Gaussian random variables, and $\{p_k(z)\}$ polynomials orthogonal on unit disk. When $p_k(z)=\sqrt{(k+1)/\pi} z^k$, $k\in \{0,1,\dots, n\}$, we give an explicit formula for $P_n(z)$ in a disk radius $r\in (0,1)$ centered at origin. From our establish limiting value ze...

A conjecture for the projection norm (or Lebesgue constant) of a weighted interpolation method based on the zeros of Chebyshev polynomials of the third and fourth kinds is resolved. This conjecture was made in a paper by J. C. Mason and G. H. Elliott in 1995. The proof of the conjecture is achieved by relating the projection norm to that of a weighted interpolation method based on zeros of Cheb...

We consider five plots of zeros corresponding to four eponymous planar polynomials (Szegő, Bergman, Faber and OPUC), for degrees up to 60, and state five conjectures suggested by these plots regarding their asymptotic distribution of zeros. By using recent results on zero distribution of polynomials we show that all these “natural” conjectures are false. Our main purpose is to provide the theor...

Let pk(x) = x +■■■ , k e N0 , be the polynomials orthogonal on [-1, +1] with respect to the positive measure da . We give sufficient conditions on the real numbers p , j = 0, ... , m , such that the linear combination of orthogonal polynomials YfLo^jPn-j has n simple zeros in (—1,-1-1) and that the interpolatory quadrature formula whose nodes are the zeros of Yfj=oßjPn-j has positive weights.