We consider real sequences (fn) that satisfy a linear recurrence with constant coefficients. We show that the density of the positivity set of such a sequence always exists. In the special case where the sequence has no positive dominating characteristic root, we establish that the density is positive. Furthermore, we determine the values that can occur as density of such a positivity set, both...

We show that the generating function (in n) for the number of walks on the square lattice with steps (1, 1), (1,−1), (2, 2) and (2,−2) from (0, 0) to (2n, 0) in the region 0 ≤ y ≤ w satisfies a very special fifth order nonlinear recurrence relation in w that implies both its numerator and denominator satisfy a linear recurrence relation.

Using a theorem of N. Chomsky and M. Schützenberger one can characterize sequences of integers which satisfy linear recurrence relations with constant coefficients (C-finite sequences) as differences of two sequences counting words in regular languages. We prove an analog for P-recursive (holonomic) sequences in terms of counting certain lattice paths.

We describe a methodology for mapping linear recurrence equations to a spectrum of systolic architectures. First, we design a systolic program in a very general architecture referred to as Basic Systolic Architecture and establish the correctness of the implementation. Next, we show how ef-cient transformations/implementations of programs for diierent systolic architectures can be obtained thro...

xn+k – qnxn = fn, n ∈N0, where k ∈N, (qn)n∈N0 , (fn)n∈N0 ⊂C, in the case qn = q, n ∈N0, q ∈C \ {0}. Using the formula, we show the existence of a unique bounded solution to the equation when |q| > 1 and supn∈N0 |fn| <∞ by finding a solution in closed form. By using the formula for the bounded solution we introduce an operator that, together with the contraction mapping principle, helps in showi...

In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur’s celebrated partition identity (1926). Andrews’ two generalisations of Schur’s theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In a recent paper, the author genera...

We describe an algorithm that takes as input a complex sequence (un) given by a linear recurrence relation with polynomial coe cients along with initial values, and outputs a simple explicit upper bound (vn) such that |un| ≤ vn for all n. Generically, the bound is tight, in the sense that its asymptotic behaviour matches that of un. We discuss applications to the evaluation of power series with...

We consider multiple orthogonal polynomials corresponding to two Macdonald functions (modified Bessel functions of the second kind), with emphasis on the polynomials on the diagonal of the Hermite-Padé table. We give some properties of these polynomials: differential properties, a Rodrigues type formula and explicit formulas for the third order linear recurrence relation. 1 Macdonald functions ...

The domination polynomial D(G, x) of a graph G is the generating function of its dominating sets. We prove that D(G, x) satisfies a wide range of reduction formulas. We show linear recurrence relations for D(G, x) for arbitrary graphs and for various special cases. We give splitting formulas for D(G, x) based on articulation vertices, and more generally, on splitting sets of vertices.

We present an algorithm for computing generators for the ideal of algebraic relations among sequences which are given by homogeneous linear recurrence equations with constant coefficients. Knowing these generators makes it possible to use Gröbner basis methods for carrying out certain basic operations in the ring of such sequences effectively. In particular, one can answer the question whether ...