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We give the asymptotic behavior of the ratio of two neighboring multiple orthogonal polynomials under the condition that the recurrence coefficients in the nearest neighbor recurrence relations converge.
We consider random trigonometric polynomials with general dependent coefficients. We show that under mild hypotheses on the structure of dependence, the asymptotics as the degree goes to infinity of the expected number of real zeros…
There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size $N$…
A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients.…
Recursive matrices are ubiquitous in combinatorics, which have been extensively studied. We focus on the study of the sums of $2\times 2$ minors of certain recursive matrices, the alternating sums of their $2\times 2$ minors, and the sums…
Many automatic sequences, such as the Thue-Morse sequence or the Rudin-Shapiro sequence, have some desirable features of pseudorandomness such as a large linear complexity and a small well-distribution measure. However, they also have some…
In a previous paper, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we present a proof for the…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
Abstract numeration systems encode natural numbers using radix ordered words of an infinite regular language and linear recurrence sequences play a key role in their valuation. Sequence automata, which are deterministic finite automata with…
In this paper, we study a class of orthogonal polynomials defined by a three-term recurrence relation with periodic coefficients. We derive explicit formulas for the generating function, the associated continued fraction, the orthogonality…
We compute the auto-correlations functions of order $m\ge 1$ for the characteristic polynomials of random matrices from certain subgroups of the unitary groups $\U(2)$ and $\U(3)$ by applying branching rules. These subgroups can be…
Schur's transforms of a polynomial are used to count its roots in the unit disk. These are generalized them by introducing the sequence of symmetric sub-resultants of two polynomials. Although they do have a determinantal definition, we…
Many polynomial invariants are defined on graphs for encoding the combinatorial information and researching them algebraically. In this paper, we introduce the cycle polynomial and the path polynomial of directed graphs for counting cycles…
The Stirling numbers of the first kind can be represented in terms of a new class of polynomials that are closely related to the Bernoulli polynomials. Recursion relations for these polynomials are given.
We show how polynomial path orders can be employed efficiently in conjunction with weak innermost dependency pairs to automatically certify polynomial runtime complexity of term rewrite systems and the polytime computability of the…
In this paper, we study the abelian complexity of the Rudin-Shapiro sequence and a related sequence. We show that these two sequences share the same complexity function $\rho(n)$ which satisfies certain recurrence relations. As a…
One relates factorization of bivariate polynomials to singularities of projective plane curves. One proves that adjoint polynomials permit to solve the recombinations of the modular factors induced by the absolute and rational…
We study the monic orthogonal polynomials with respect to a singularly perturbed Airy weight. By using Chen and Ismail's ladder operator approach, we derive a discrete system satisfied by the recurrence coefficients for the orthogonal…
Two results on palindromicity of bi-infinite words in a finite alphabet are presented. The first is a simple, but efficient criterion to exclude palindromicity of minimal sequences and applies, in particular, to the Rudin-Shapiro sequence.…
A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of…