Dynamics of the zeros of analytic continued polynomials and differential equations associated with q-tangent polynomials

Differential Equations and Zeros of Orthogonal Polynomials

This is a survey of some methods for finding inequalities, monotonicity properties and approximations for zeros of orthogonal polynomials and related functions. The methods are based on the use of the ordinary differential equations satisfied by the functions. Based on a talk presented at the NATO Advanced Study Institute on Orthogonal Polynomials and their Applications, Ohio State University, ...

Fractional differential equations with Legendre polynomials

q-Eulerian Polynomials and Polynomials with Only Real Zeros

Let f and F be two polynomials satisfying F (x) = u(x)f(x) + v(x)f (x). We characterize the relation between the location and multiplicity of the real zeros of f and F , which generalizes and unifies many known results, including the results of Brenti and Brändén about the q-Eulerian polynomials.

Some Identities on the q-Tangent Polynomials and Bernstein Polynomials

In this paper, we investigate some properties for the q-tangent numbers and polynomials. By using these properties, we give some interesting identities on the q-tangent polynomials and Bernstein polynomials. Throughout this paper, let p be a fixed odd prime number. The symbol, Zp, Qp and Cp denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic ...

Some compact generalization of inequalities for polynomials with prescribed zeros

‎Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial‎ ‎of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$‎. ‎In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$‎, ‎$k^2 leq rRleq R^2$ and for $Rleq r leq k$‎. ‎Our results refine and generalize certain well-known polynomial inequalities‎.