We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for cer...

This paper deals with monic orthogonal polynomial sequences (MOPS in short) generated by a Geronimus canonical spectral transformation of a positive Borel measure μ, i.e., 1 (x− c) dμ(x) +Nδ(x− c), for some free parameter N ∈ R+ and shift c. We analyze the behavior of the corresponding MOPS. In particular, we obtain such a behavior when the mass N tends to infinity as well as we characterize th...

This paper aims to clarify certain aspects of the relations between birth-death processes, measures solving a Stieltjes moment problem, and sets of parameters defining polynomial sequences that are orthogonal with respect to such a measure. Besides giving an overview of the basic features of these relations, revealed to a large extent by Karlin and McGregor, we investigate a duality concept for...

We investigate the approximation properties of trigonometric polynomials and prove some direct and inverse theorems for polynomial approximation in weighted rearrangement invariant spaces.

We survey and present some new results that are related to the behavior of orthogonal polynomials in the plane under small perturbations of the measure of orthogonality. More precisely, we introduce the notion of a polynomially small (PS) perturbation of a measure. Namely, if μ0 ≥ μ1 and {pn(μj , z)}n=0, j = 0, 1, are the associated orthonormal polynomial sequences, then μ0 a PS perturbation of...

We study problems of reconfiguration of shortest paths in graphs. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length. Moreover, we also study reconfiguration of independent sets in three different models and analyze relati...

Let f(z) 6≡ 0 be a solution of f ′′ + P (z)f = 0, where P (z) is a polynomial. Then the set of accumulation lines of zero-sequence is a subset of the Borel directions of f(z). Let f1 and f2 be two linearly independent solutions of f ′′ + P (z)f = 0, where P (z) is a polynomial of degree n and set E = f1f2. Then, for every accumulation line arg z = θ of zero-sequence of E, there is another accum...

Consider sequences of integers {Un}TM=0 defined by U„ = aUn_x +bUn_2 for all integers n > 2, where U0 = 0, C/j = 1, a and b are given integers. We call these sequences generalized Fibonacci sequences with parameters a and b. In the case where a = b = 1, the sequence {Un}TM=0 is called the Fibonacci sequence, and we denote its terms by F0,Fl9.... The polynomial f(x) = x-ax-h with discriminant D ...

Remarkable breakthroughs have been made recently in obtaining approximate solutions to some fundamental NP-hard problems, namely Max-Cut, Max k-Cut, Max-Sat, Max-Dicut, Max-bisection, k-vertex coloring, maximum independent set, etc. All these breakthroughs involve polynomial time randomized algorithms based upon semidefinite programming, a technique pioneered by Goemans and Williamson. In this ...

In this article, we establish polynomial-growth bound for the sequence of Fourier coefficients associated with even integer weight vector-valued automorphic forms non-cocompact Fuchsian groups first kind. end, their L-functions and exponential sums have been discussed.