In this paper, we define a new form of Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials by generalizing the Euler polynomials, q-Euler polynomials. Some interesting identities, explicit formulas, symmetric properties, connection with are obtained. Finally, investigate zeros using computer.

We derive new conditions for nonexistence of integral zeros of binary Krawtchouk polynomials. Upper bounds for the number of integral roots of Krawtchouk polynomials are presented.

∞ n=0 pn(x)t n or, equivalently, p′n(x) = pn−1(x). If g(t) is an entire function, g(0) 6= 0, with at least one zero, the asymptotics of linearly scaled polynomials {pn(nx)} are described by means of finitely zeros of g, including those of minimal modulus. As a consequence, we determine the limiting behavior of their zeros as well as their density. The techniques and results extend our earlier w...

FINE STRUCTURE OF THE ZEROS OF ORTHOGONAL POLYNOMIALS, I. A TALE OF TWO PICTURES BARRY SIMON Dedicated to Ed Saff on the occasion of his 60th birthday Abstract. Mhaskar-Saff found a kind of universal behavior for the bulk structure of the zeros of orthogonal polynomials for large . Motivated by two plots, we look at the finer structure for the case of random Verblunsky coefficients and for what...

The purpose of this paper is to present three new methods for finding all simple zeros of polynomials simultaneously. First, we give a new method for finding simultaneously all simple zeros of polynomials constructed by applying the Weierstrass method to the zero in the trapezoidal Newton’s method, and prove the convergence of the method. We also present two modified Newton’s methods combined w...

We study the variation of the zeros of the Hermite function Hλ(t) with respect to the positive real variable λ. We show that, for each nonnegative integer n, Hλ(t) has exactly n + 1 real zeros when n < λ ≤ n + 1 and that each zero increases from −∞ to ∞ as λ increases. We establish a formula for the derivative of a zero with respect to the parameter λ; this derivative is a completely monotonic ...

We show that the variance of the number of simultaneous zeros ofm i.i.d. Gaussian random polynomials of degree N in an open set U ⊂ C with smooth boundary is asymptotic to N νmm Vol(∂U), where νmm is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on C. Our res...

The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This is the first density result for the zeros of the Tutte polynomial in a region of positive volume. Our res...

The Newton polytope Pf of a polynomial f is well known to have a strong impact on its behavior. The Kouchnirenko-Bernstein theorem asserts that even the number of simultaneous zeros in (C∗)m of a system of m polynomials depends on their Newton polytopes. In this article, we show that Newton polytopes further have a strong impact on the distribution of mass and zeros of polynomials, the basic th...

We use Turán type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term recurrence pk+1 = xpk − ckpk−1, with a nondecreasing sequence {ck}. As a special case they include a non-asymptotic version of Máté, Nevai and Tot...