A sequence of complex numbers is holonomic if it satisfies a linear recurrence with polynomial coefficients. A power series is holonomic if it satisfies a linear differential equation with polynomial coefficients, which is equivalent to its coefficient sequence being holonomic. It is well known that all algebraic power series are holonomic. We show that the analogous statement for sequences is ...

A nonhomogeneous system of linear recurrence equations can be recognized by an automaton A over a one-letter alphabet A = {z}. Conversely, the automaton A generates precisely this nonhomogeneous system of linear recurrence equations. We present the solutions of these systems and apply the z-transform to these solutions to obtain their series representation. Finally, we show some results that si...

Global asymptotic stability of rational difference equations is an area of research that has been well studied. In contrast to the many current methods for proving global asymptotic stability, we propose an algorithmic approach. The algorithm we summarize here employs the idea of contractions. Given a particular rational difference equation, defined by a function Q : R → R, we attempt to find a...

We introduce a new family of sequences {tk(n)}n=−∞ for given positive integer k > 3. We call these new sequences as generalized Alcuin’s sequences because we get Alcuin’s sequence which has several interesting properties when k = 3. Also, {tk(n)}n=0 counts the number of partitions of n − k with parts being k, (k − 1), 2 (k − 1), 3 (k − 1), . . . , (k − 1) (k − 1). We find an explicit linear rec...

We introduce a novel fourth order linear recurrence sequence {Sn} using the two periodic binary recurrence. We call it “pellans sequence” and then we solve the system ab+ 1 = Sx, ac + 1 = Sy bc+ 1 = Sz where a < b < c are positive integers. Therefore, we extend the order of recurrence sequence for this variant diophantine equations by means of pellans sequence.

We formulate a linear difference equation which yields averaged semi-inclusive decay rates for arbitrary, not necessarily large, values of the masses. We show that the rates for decays M → m+M ′ of typical heavy open strings are independent of the masses M and m, and compute the “mass deffect” M−m−M ′. For closed strings we find decay rates proportional to Mm(1−D)/2 R , wheremR is the reduced m...

The main goal of the paper is to investigate boundedness, invariant intervals, semicycles, and global attractivity of all nonnegative solutions of the equation xn 1 α βxn γxn−k / 1 xn−k , n ∈ N0, where the parameters α, β, γ ∈ 0,∞ , k ≥ 2 is an integer, and the initial conditions x−k, . . . , x0 ∈ 0,∞ . It is shown that the unique positive equilibrium of the equation is globally asymptotically ...

In this note, we study the general form of some rational recursive sequences. By some modification of the methods and ideas, as well as the transformation from the paper [K. The global attractivity of the rational difference equation y n = y n−k +yn−m 1+y n−k yn−m , Appl. Math. Lett. 20 (2007), 54–58], we give a new proof for the conjectures posed therein.

whenever (un) ∞ n=0 is a non-degenerate linear recurrence sequence. Mahler’s proof is not effective in the following sense. Given a positive integer m the proof does not yield a number C(m) which is effectively computable in terms of m, such that |un| > m whenever n > C(m). However, Schmidt [31, 32], Allen [1], and Amoroso and Viada [2] have given estimates in terms of t only for the number of ...

The diversity of life is a fundamental empirical fact of nature. Casual observation alone confirms the great variety of species. But diversity also prevails within single species. As in human populations, the individuals of a species vary considerably in their outward traits—size and shape, external markings, fecundity, disease resistance, etc. This is called phenotypic variation, and, since ph...