Schauder's fixed point technique is applied to asymptotical analysis of solutions of a linear Volterra difference equation xðn þ 1Þ ¼ aðnÞ þ bðnÞxðnÞ þ P n i¼0 Kðn; iÞxðiÞ In the paper, sufficient conditions are derived for the validity of a property of solutions that, for every admissible constant c 2 R, there exists a solution x ¼ xðnÞ such that xðnÞ $ c þ P nÀ1 i¼0 aðiÞ bði þ 1Þ bðnÞ; where ...

This note compares and contrasts some properties of binary sequences with matrices and associated recurrence relations in order to stimulate some enrichment exercises and pattern puzzles.

We prove that the question of whether a given linear partial differential or difference equation with polynomial coefficients has non-zero polynomial solutions is algorithmically undecidable. However, for equations with constant coefficients this question can be decided very easily since such an equation has a non-zero polynomial solution iff its constant term is zero. We give a simple combinat...

We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of three-term linear recurrence relations we show that there exists a four-term linear recurrence relation whose solutions allow us to show that the number is a quadratic irrational if and only if the four-term ...

We formulate a result that states that specific products of two independent solutions of a real three-term recurrence relation will form a basis for the solution space of a four-term linear recurrence relation (thereby extending an old result of Clausen [7] in the continuous case to this discrete setting). We then apply the theory of disconjugate linear recurrence relations to the study of irra...

We solve an eigenvalue equation that appears in several papers about a wide range of physical problems. The Frobenius method leads to three-term recurrence relation for the coefficients power series that, under suitable truncation, yields exact analytical eigenvalues and eigenfunctions particular values model parameter. From these solutions, some researchers have derived variety predictions suc...

In this work we provide low rank estimations for coordinate sequences of linear recurrent sequences (LRS) of maximal period (MP) over Galois ring R = GR(pn, r), p ≥ 5, r ≥ 2, with numbers s such that s = kr + 2, k ∈ N0.

We derive recurrence relations between phase space expressions in different dimensions by confining some of the coordinates to tori or spheres of radius R and taking the limit as R → ∞. These relations take the form of mass integrals, associated with extraneous momenta (relative to the lower dimension), and produce the result in the higher dimension.

Orthogonal zero interpolants (OZI) are polynomials which interpolate the “zero-function” at a finite number of pre-assigned nodes and satisfy orthogonality condition. OZI’s can be constructed by the 3-term recurrence relation. These interpolants are found useful in the solution of constrained approximation problems and in the structure of Gauss-type quadrature rules. We present some theoretical...

The 3-term recurrence relation for Hermite polynomials was recently generalized to a Wronskians of polynomials. In this note, similar generalization Laguerre is obtained.