In the solution of the superintegrable chiral Potts model special polynomials related to the representation theory of the Onsager algebra play a central role. We derive approximate analytic formulae for the zeros of particular polynomials which determine sets of low-lying energy eigenvalues of the chiral Potts quantum chain. These formulae allow the analytic calculation of the leading finite-si...

We investigate multiple Charlier polynomials and in particular we will use the (nearest neighbor) recurrence relation to find the asymptotic behavior of the ratio of two multiple Charlier polynomials. This result is then used to obtain the asymptotic distribution of the zeros, which is uniform on an interval. We also deal with the case where one of the parameters of the various Poisson distribu...

We unify and generalize several known results about systems of random polynomials. We first classify all orthogonally invariant normal measures for spaces of polynomial mappings. For each such measure we calculate the expected number of real zeros. The results for invariant measures extend to underdetermined systems, giving the expected volume for orthogonally invariant random real projective v...

The Fibonacci polynomials are defined by the recursion relation Fn+2{x) = xF„+l(x) + Fn(x), (1) with the initial values Fx(x) = 1 and F2(x) = x. When x = l, Fn(x) is equal to the /1 Fibonacci number, Fn. The Lucas polynomials, Ln(x) obey the same recursion relation, but have initial values Li(x) = x and L^x) = x +2. Explicit expressions for the zeros of the Fibonacci and Lucas polynomials have ...

We consider polynomials on the unit circle defined by the recurrence relation Φk+1(z) = zΦk(z)− αkΦk(z) k ≥ 0, Φ0 = 1 For each n we take α0, α1, . . . , αn−2 i.i.d. random variables distributed uniformly in a disk of radius r < 1 and αn−1 another random variable independent of the previous ones and distributed uniformly on the unit circle. The previous recurrence relation gives a sequence of ra...

We consider zeros of polynomials whose coefficients lie in the field C((t)) of formal Laurent series with complex coefficients. The algebraic closure of C((t)) is the field K of “Puiseux series,” which allow fractional exponents. There is a well-known algorithm, described in [9], for constructing roots in K to a one-variable polynomial over C((t)). Several papers give generalizations of this al...

We study the average number of intersecting points of a given curve with random hyperplanes in an n-dimensional Euclidean space. As noticed by A. Edelman and E. Kostlan this problem is closely linked to nding the average number of real zeros of random polynomials. They show that a real polynomial of degree n has in average 2 log n + O(1) real zeros (M. Kac's theorem). This result leads us to th...

In this paper, we study polynomials with only real zeros based on the method of compatible zeros. We obtain a necessary and sufficient condition for the compatible property of two polynomials whose leading coefficients have opposite sign. As applications, we partially answer a question proposed by M. Chudnovsky and P. Seymour in the recent publication [M. Chudnovsky, P. Seymour, The roots of th...

In this paper we study the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. The zeros of these polynomials are the nodes for complex Gaussian quadrature of an oscillatory integral on the real axis with a high order stationary point, and their limit distribution is also analyzed. We show that the z...

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between con...