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A new bidimensional version of cobalancing numbers and Lucas-balancing numbers are introduced. Some properties and identities satisfied by these new bidimensional sequences are studied. [ABSTRACT FROM AUTHOR]

During the conversion from difference to differential in the grey polynomial model, the "misplaced replacement" problem will occur. A novel unbiased grey polynomial model, i.e., HUGMP(1, 1,N) is presented to overcome the above drawback. Meanwhile, the parameter estimation of HUGMP (1, 1,N) is directly constructed by the equivalent relation between the parameter estimation and the recurrence relation of the time response function for GMP(1, 1,N) model. The recurrence relation is deduced from the solution of the homogenized differential equations, converted from the whitenization equation of GMP(1, 1,N) model by introducing new variables. The simulated values are directly calculated by the precise direct integration method in order to reduce round-off error and improve fitting accuracy. Moreover, it is proved that the proposed unbiased grey polynomial model possesses not only complete coincidence of simulation to non-homogeneous exponential sequence with polynomial time terms, but also multiple transformation consistency. At last, the results of applications verify the effectiveness of the proposed model by comparing with other conventional models. [ABSTRACT FROM AUTHOR]

Suppose that a 0 , a 1 , ... is an integer sequence which satisfies a recurrence relation with constant coefficients, and let T (x) = f (x) / g (x) be its generating function, where f (x) and g (x) have no common factors in Z [ x ]. In this article, we study the problem of finding the rational values of x such that T (x) is an integer. We say that such a number is good for the sequence. Our first main result is that if g (x) has at least two different irreducible factors, or if g (x) has a single irreducible factor of degree at least 3, then the sequence has only finitely many good values. We also study sequences of the form 0 , 1 , ... for which the recurrence relation has order 2. Among other results, we show that under a mild condition on the recurrence relation, the sequence has infinitely many good values, and we give a constructive method to find all of them. [ABSTRACT FROM AUTHOR]

This is an expansion and modiÖcation of the paper from [6]. We discuss the convergence of locus in the paper [5], which originated from a practice problem for the Chinese college entrance exam. Next, we extended some results in [6] from 2D to 3D. We are interested in the limit of a recursive sequence of loci that is built on a special convex combination of vectors involving curves or surfaces. We shall see many interesting graphs of uniform convergence of sequences generated by parametric curves and surfaces, which we hope to inspire many applications in computer graphics, and other related disciplines. [ABSTRACT FROM AUTHOR]

Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko [SIAM J. Discrete Math. 3 (1990), pp. 216–240], except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes). [ABSTRACT FROM AUTHOR]

Abstract. We construct new (n + 1)-variable models of irreducible representations of the Lie algebra K5. An n-fold integral transformation is used to obtain a new set of models of K5 in terms of difference-differential operators. These models are further exploited to obtain recurrence relations, generating functions and addition theorems. [ABSTRACT FROM AUTHOR]

This article discusses the topic of shifted Fibonacci numbers. The author explores the Binet formula, which is used to find the necessary elements of the shifted Fibonacci number sequence. They also present various identities and summation formulas for this sequence. The article fills a gap in the literature and provides a foundation for future research on additional identities or generalizations for shifted Fibonacci numbers. [Extracted from the article]

Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree d whose Galois group is $S_d$. Let $(a_n)$ be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence $(a_n)$ is positive. [ABSTRACT FROM AUTHOR]

Let (U_n)_{n\in \mathbb {N}} be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants B and N_0 such that for any b,c\in \mathbb {Z} with b> B the equation U_n - b^m = c has at most two distinct solutions (n,m)\in \mathbb {N}^2 with n\geq N_0 and m\geq 1. Moreover, we apply our result to the special case of Tribonacci numbers given by T_1= T_2=1, T_3=2 and T_{n}=T_{n-1}+T_{n-2}+T_{n-3} for n\geq 4. By means of the LLL-algorithm and continued fraction reduction we are able to prove N_0=2 and B=e^{438}. The corresponding reduction algorithm is implemented in Sage. [ABSTRACT FROM AUTHOR]

We consider recursive sequences over the set of integers, where as rules of generation we take arbitrary superpositions of polynomial functions and the function |x|; such sequences are referred to as polynomial-modular recursive sequences. We show how evaluations on three-tape Minsky machines can be simulated via polynomial-modular recursive sequences. Based on this result, we formulate algorithmically unsolvable problems related to polynomial-modular recursive sequences. We also consider recursive sequences in which the rules of generation are functions formed by some superpositions of polynomial functions and the function [ x ]. For the set of such recursive sequences, an algorithmically unsolvable problem is indicated. [ABSTRACT FROM AUTHOR]

The set of indices that correspond to the positive entries of a sequence of numbers is called its positivity set. In this paper, we study the density of the positivity set of a given linear recurrence sequence, that is the question of how much more frequent are the positive entries compared to the non-positive ones. We show that one can compute this density to arbitrary precision, as well as decide whether it is equal to zero (or one). If the sequence is diagonalisable, we prove that its positivity set is finite if and only if its density is zero. Further, arithmetic properties of densities are treated, in particular we prove that it is decidable whether the density is a rational number, given that the recurrence sequence has at most one pair of dominant complex roots. Finally, we generalise all these results to symbolic orbits of linear dynamical systems, thereby showing that one can decide various properties of such systems, up to a set of density zero. [ABSTRACT FROM AUTHOR]

In this paper, we use the method of error estimation to consider the reciprocal sums of products of any m th-order linear recurrence sequences { u n } . Specifically, we find that a series of sequences are "asymptotically equivalent" to the reciprocal sums of products of any m th-order linear recurrence sequences { u n } . [ABSTRACT FROM AUTHOR]

In this paper we explore some of the geometry that lies behind the real linear, second order, constant coefficient, recurrence relation (1) where a and b are real numbers. Readers will be familiar with the standard method of solving this relation, and, to avoid trivial cases, we shall assume that ab ≠ 0. The auxiliary equation of t 2 = at + b of (1) has two (possibly complex) solutions and the most general solution of (1) is given by (i) when are real and distinct; (ii) when (iii) . [ABSTRACT FROM AUTHOR]

In this paper, closed forms of the sum formulas ∑nk=0kxkWk and ∑nk=1kxkW-k for generalized Hexanacci numbers are presented. As special cases, we give summation formulas of Hexanacci, Hexanacci-Lucas, and other sixth-order recurrence sequences. [ABSTRACT FROM AUTHOR]

An effective boundary condition on a porous wall is derived, starting from basic principles of mechanics. Stokes system, governing the viscous flow through a reservoir with an array of small pores on the boundary, was studied, and the corresponding macroscopic model via rigorous asymptotic analysis is found. Under the assumption of periodicity of the pores, the effective boundary condition of the Darcy type is derived, using homogenization and boundary layer techniques. Further asymptotic analysis with respect to the porosity yields a recursive sequence of boundary value problems showing that the large pressure jump occurs on the boundary. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 . A generalization of the well-known Pell sequence is the k-Pell sequence. For this sequence, the first k terms are 0 , ... , 0 , 1 and each term afterwards is given by the linear recurrence P n (k) = 2 P n - 1 (k) + P n - 2 (k) + ⋯ + P n - k (k). In this paper, we extend the previous work (Rihane and Togbé in Ann Math Inform 54:57–71, 2021) and investigate the Padovan and Perrin numbers in the k-Pell sequence. [ABSTRACT FROM AUTHOR]

We study recurrence relations satisfied by the moments λn (z) of discrete linear functionals whose first moment satisfies a holonomic differential equation. We consider all cases when the order of the ODE is less or equal than 3. [ABSTRACT FROM AUTHOR]

In this paper, we study the form of the solution of the following systems of difference equations of order two ... with nonzero real numbers initial conditions. [ABSTRACT FROM AUTHOR]

If a sequence indexed by nonnegative integers satisfies a linear recurrence without constant terms, one can extend the indices of the sequence to negative integers using the recurrence. Recently, Cigler and Krattenthaler showed that the negative version of the number of bounded Dyck paths is the number of bounded alternating sequences. In this paper, we provide two methods to compute the negative versions of sequences related to moments of orthogonal polynomials. We give a combinatorial model for the negative version of the number of bounded Motzkin paths. We also prove two conjectures of Cigler and Krattenthaler on reciprocity between determinants. [ABSTRACT FROM AUTHOR]

Recently, Chappelon and Ramírez Alfonsín defined the square Frobenius number of coprime numbers m and n to be the largest perfect square that cannot be expressed in the form mx + ny for nonnegative integers x and y. When m and n differ by 1 or 2, they found simple expressions for the square Frobenius number if neither m nor n is a perfect square. If either m or n is a perfect square, they formulated some interesting conjectures which have an unexpected close connection with a known recursive sequence, related to the denominators of Farey fraction approximations to √ 2. In this note, we prove these conjectures. Our methods involve solving Pell equations x² - 2y² = 1 and x² - 2y² = -1. Finally, to complete our proofs of these conjectures, we eliminate several cases using several results related to solutions of simultaneous Pell equations. [ABSTRACT FROM AUTHOR]

We provide a novel recursive method, which does not require any assumption, to compute the entries of the kth power of a semicirculant matrix by using a closed-form expression for the mth term of a certain recursive polynomial sequence. As an application, a method for computing the entries of the kth power of r-circulant matrices is also presented. [ABSTRACT FROM AUTHOR]

Let S = { p 1 , ... , p s } be a finite, non-empty set of distinct prime numbers and (U n) n ≥ 0 be a linear recurrence sequence of integers of order at least 2. For any positive integer k, and w = (w k , ... , w 1) ∈ Z k , w 1 , ... , w k ≠ 0 we define (U j (k , w)) j ≥ 1 an increasing sequence composed of integers of the form | w k U n k + ⋯ + w 1 U n 1 | , n k > ⋯ > n 1 . Under certain assumptions, we prove that for any ε > 0 , there exists an integer n 0 such that [ U j (k , w) ] S < (U j (k , w)) ε , for j > n 0 , where [ m ] S denotes the S-part of the positive integer m. On further assumptions on (U n) n ≥ 0 , we also compute an effective bound for [ U j (k , w) ] S of the form (U j (k , w)) 1 - c , where c is a positive constant depending only on r, a 1 ,..., a r , U 0 ,..., U r - 1 , w 1 ,..., w k and S. [ABSTRACT FROM AUTHOR]

We apply general moment identities for Poisson stochastic integrals with random integrands to the computation of the moments of Markovian growth–collapse processes. This extends existing formulas for mean and variance available in the literature to closed-form moment expressions of all orders. In comparison with other methods based on differential equations, our approach yields explicit summations in terms of the time parameter. We also treat the case of the associated embedded chain, and provide recursive codes in Maple and Mathematica for the computation of moments and cumulants of any order with arbitrary cut-off moment sequences and jump size functions. [ABSTRACT FROM AUTHOR]

Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward descriptions in terms of how often each value in the sequence occurs. In this paper, we generalize the most classical examples to a larger family of sequences parametrized by linear recurrence relations. Each of our sequences can be constructed in three different ways: via a nested recurrence relation, from labeled infinite trees, or by using Zeckendorf-like strings of digits to describe its frequency sequence. We conclude the paper by discussing the asymptotic behaviors of our sequences. [ABSTRACT FROM AUTHOR]

For an integer q ≥ 2 , a q-recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of q. In this article, q-recursive sequences are studied and the asymptotic behavior of their summatory functions is analyzed. It is shown that every q-recursive sequence is q-regular in the sense of Allouche and Shallit and that a q-linear representation of the sequence can be computed easily by using the coefficients from the recurrence relations. Detailed asymptotic results for q-recursive sequences are then obtained based on a general result on the asymptotic analysis of q-regular sequences. Three particular sequences are studied in detail: We discuss the asymptotic behavior of the summatory functions of Stern's diatomic sequence, the number of non-zero elements in some generalized Pascal's triangle and the number of unbordered factors in the Thue–Morse sequence. For the first two sequences, our analysis even leads to precise formulæ without error terms. [ABSTRACT FROM AUTHOR]

A simple model of the new notion of "Markov up" processes is proposed; its positive recurrence and ergodic properties are shown under the appropriate conditions. A one-dimensional process in discrete time moves upwards as if it were Markov, and goes down in a more complicated way, remembering all its past from the moment of its "u-turn" down. Also, it is assumed that in some sense its move downwards becomes more and more probable after each step in this direction. [ABSTRACT FROM AUTHOR]

In this paper, we derive some identities and recurrence relations for the degenerate Stirling numbers of the first kind and of the second kind, which are degenerate versions of the ordinary Stirling numbers of the first kind and of the second kind. They are deduced from the normal orderings of degenerate integral powers of the number operator and their inversions, certain relations of boson operators and the recurrence relations of the Stirling numbers themselves. Here we note that, while the normal ordering of an integral power of the number operator is expressed with the help of the Stirling numbers of the second kind, that of a degenerate integral power of the number operator is represented by means of the degenerate Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

In the present work, two new recurrences of the Padovan sequence given with delayed initial conditions are defined. Some identities of these sequences which we call the Padovan arrays were examined. Also, generating and series functions of the Padovan arrays are examined. [ABSTRACT FROM AUTHOR]

In this paper, we define triangular Pell and triangular Pell-Lucas numbers. We carry with little differences an elegant result given in the literature on the sum of any two consecutive triangular numbers from triangular numbers to triangular Pell numbers and Pell-Lucas numbers. Also, we present some interesting identities satisfied by the triangular Pell and triangular Pell-Lucas numbers which are connected with Pell and Pell-Lucas numbers. Furthermore, we give some important properties of triangular Pell and triangular Pell-Lucas numbers. [ABSTRACT FROM AUTHOR]

The core of a finite-dimensional modular representation M of a finite group G is its largest non-projective summand. We prove that the dimensions of the cores of M ⊗ n have algebraic Hilbert series when M is Omega-algebraic, in the sense that the non-projective summands of M ⊗ n fall into finitely many orbits under the action of the syzygy operator Ω. Similarly, we prove that these dimension sequences are eventually linearly recursive when M is what we term Ω + -algebraic. This partially answers a conjecture by Benson and Symonds. Along the way, we also prove a number of auxiliary permanence results for linear recurrence under operations on multi-variable sequences. [ABSTRACT FROM AUTHOR]

Our work focuses on the transition from high school to university in the field of calculus. In France, recursive sequences are studied as one of the classical exercises in both institutions. Their studies use different theorems and notions, such as functions, convergence, monotonicity, induction, etc. The work expected at this transition requires the development of recognition and control activities on the part of the students. We propose a new task allowing the development of such activities from a dialectic between two paradigms of analysis using the calculator. We highlight that students at the end of high school have difficulties in studying these sequences, and they do not easily develop the activities of recognition and control independently. This could cause problems in understanding recursive sequences at the beginning of university. [ABSTRACT FROM AUTHOR]

K.T. Atanassov are Firstly established the Coupled Fibonacci Sequence in 1985. In 1987, The essence of Fibonacci Triple Sequences are examined. Fibonacci Sequence stand out as a kind of super sequence with amazing properties. This is the meteoric expansion in the province of Fibonacci Sequence. Leonardo de Pisa foremost Fibonacci's observation on the growth of the rabbit population as a result in 1202. Triple Fibonacci Sequence are hype in the last years, but Multiplicative Triple Sequence of Recurrence Relations are less known. Extravagant work has been done to course on Fibonacci Triple Sequence in Additive form. In 1995, Multiplicative Coupled Fibonacci Sequence are treated. Our wish of this paper to offer some results of Multiplicative Triple Fibonacci Sequence of fourth order under nine specific schemes. [ABSTRACT FROM AUTHOR]

We address the following decision problem. Given a numeration system U and a U-recognizable set X ⊆ ℕ , i.e. the set of its greedy U-representations is recognized by a finite automaton, decide whether or not X is ultimately periodic. We prove that this problem is decidable for a large class of numeration systems built on linear recurrence sequences. Based on arithmetical considerations about the recurrence equation and on p-adic methods, the DFA given as input provides a bound on the admissible periods to test. [ABSTRACT FROM AUTHOR]

A differentially recursive sequence over a differential field is a sequence of elements satisfying a homogeneous differential equation with non-constant coefficients (namely, Taylor expansions of elements of the field) in the differential algebra of Hurwitz series. The main aim of this paper is to explore the space of all differentially recursive sequences over a given field with a non-zero differential. We show that these sequences form a two-sided vector space that admits, in a canonical way, a structure of Hopf algebroid over the subfield of constant elements. We prove that it is the direct limit, as a left comodule, of all spaces of formal solutions of linear differential equations and that it satisfies, as Hopf algebroid, an additional universal property. When the differential on the base field is zero, we recover the Hopf algebra structure of linearly recursive sequences. [ABSTRACT FROM AUTHOR]

In this study, one-dimensional, two-dimensional, three-dimensional and n-dimensional recurrences of the (p; q)-Fibonacci sequence are examined and their some identities are given. [ABSTRACT FROM AUTHOR]

In this work, we aim to study some qualitative properties of higher order nonlinear difference equations. Specifically, we investigate local as well as global stability and boundedness of solutions of this equation. In addition, we will provide solutions to a number of special cases of the studied equation. Also, we present many numerical examples that support the results obtained. The importance of the results lies in completing the results in the literature, which aims to develop the theoretical side of the qualitative theory of difference equations. [ABSTRACT FROM AUTHOR]

It is tempting to accept bets when the outcome has a positive expectation favouring the bettor. We examine situations where this is not enough of a criterion as a basis for betting. The argument we present shows that the probability of winning the bets in the long run has to be qualified in a certain way beyond positive expectation. We introduce an alternative index as a criterion to recommend to bettors. [ABSTRACT FROM AUTHOR]

Most nonlinear difference equations have exact solutions that are not always possible to obtain theoretically. As a result, a large number of researchers investigate several qualitative aspects of difference equations in order to predict their lengthy behavior. The goal of our research is to obtain the solutions of a tenth-order difference equation U n + 1 = U n − 9 U n − 5 U n − 1 / U n − 7 U n − 3 ± 1 ± U n − 9 U n − 5 U n − 1 , n ≥ 0 , where the initial values are positive real numbers. Stability and periodicity are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we construct and study Rauzy partitions of order n for a certain class of Pisot numbers. These partitions are partitions of a torus into fractal sets. Moreover, the action of a certain shift of the torus on partitions introduced is reduced to rearranging the partition tiles. We obtain a number of applications of partitions introduced to the study of the corresponding shift of the torus. In particular, we prove that partition tiles are sets of bounded remainder with respect to the shift considered. In addition, we obtain a number of applications to the study of sets of positive integers that have a given ending of the greedy expansion by a linear recurrent sequence and to generalized Knuth–Matiyasevich multiplications. [ABSTRACT FROM AUTHOR]

The present work presents some examples of data originated from a set of investigations developed in Brazil, at the Federal Institute of Education, Science and Technology of the State of Ceará - IFCE, in the context of teacher education, and the consideration of elements of a historical, mathematical, and evolutionary nature. Thus, from a perspective of theoretical complementarity involving the research elements derived from the notion of the didactic engineering for development and some assumptions of the French aspect of the professional didactics, two sets of data are discussed. The first set involved research carried out between 2017 and 2020. The second set considered involved studies still under development in Brazil, considering the period of 2020 - 2023. The historical landscape of investigation rests on considering the recurring number sequences of Fibonacci, Lucas, Pell, Jacobsthal, Coordonier or Padovan, Perrin, Mersenne, Oresme, Naraynna, and Leonardo. Furthermore, in a broad sense, the study aims to provide a learning scenario for the teacher, affected by a perspective of mathematical knowledge evolution, including the repercussions and applications with current technology. [ABSTRACT FROM AUTHOR]

Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class In(=, 6=,>) is the same as that of n - 1 - asc on the class In(>, 6=,=), which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on In(=, 6=,>) of asc and desc along with two other parameters, and do the same for n - 1 - asc and desc on In(>, 6=,=). By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result that the common cardinality of In(=, 6=,>) and In(>, 6=,=) is the same as that of Sn(4231, 42513). In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents. [ABSTRACT FROM AUTHOR]

Initiated by Davis, Nelson, Petersen and Tenner (2018), the enumerative study of pinnacle sets of permutations has attracted a fair amount of attention recently. In this article, we provide a recurrence that can be used to compute efficiently the number jSn(P)j of permutations of size n with a given pinnacle set P, with arithmetic complexity O(k4 + k log n) for P of size k. A symbolic expression can also be computed in this way for pinnacle sets of fixed size. A weighted sum qn(P) of jSn(P)j proposed in Davis, Nelson, Petersen and Tenner (2018) seems to have a simple form, and a conjectural form is given recently by Flaque, Novelli and Thibon (2021+). We settle the problem by providing and proving an alternative form of qn(P), which has a strong combinatorial flavor. We also study admissible orderings of a given pinnacle set, first considered by Rusu (2020) and characterized by Rusu and Tenner (2021), and we give an efficient algorithm for their counting. [ABSTRACT FROM AUTHOR]

In this paper, closed forms of the sum formulas ∑k=1n GWk², ∑k=1n GWk+2GWk and ∑k=1n GWk+1GWk for the squares of Gaussian generalized Tribonacci numbers are presented. As special cases, we give sum formulas of Gaussian Tribonacci, Gaussian Tribonacci-Lucas, Gaussian Padovan, Gaussian Perrin, Gaussian Narayana and some other third order linear recurrence sequences. All the summing formulas of well known recurrence sequences are linear except the cases Gaussian Pell-Padovan and Gaussian Padovan-Perrin. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the t-extension of the p-Fibonacci matrix and give a Factorization of the Pascal matrix involving the t-extension of the p-Fibonacci matrix. Also, we obtain some results on the relations between the Stirling matrix of the second kind and the 1-extension of the p-Fibonacci matrix. [ABSTRACT FROM AUTHOR]

In this paper, the exact expressions as well as recurrence relations for single and product moments of the generalized lower record values from generalized inverse Weibull distribution are obtained. Further, the characterization of the given distribution is carried out through recurrence relations and conditional moment. [ABSTRACT FROM AUTHOR]

Termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination relates to deep open problems in number theory, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this article, we introduce the class of o-minimal invariants, which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel's conjecture is transcendental number theory. [ABSTRACT FROM AUTHOR]

We give combinatorial interpretations of several sequences defined recurrently in terms of restricted words over a finite alphabet. One of the main tools for such investigations is the notion of invert transform which allows us to enlarge the alphabet by one letter. The initial sequence f0 is defined via a linear homogeneous recurrence of the second order. Then, we define, for each integer n ≥ 1, the sequence fn as the invert transform of fn-1. For a number of such recurrences we find an explicit formula for its solutions as well as their interpretations in terms of restricted words. Explicit bijections between different sets of restricted words counted by the same Fibonacci number are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we show that if (X n , Y n) is the n th solution of the Pell equation X 2 − d Y 2 = ± 1 for some non-square d , then given any integer c , the equation c = X n − 2 m has at most 2 integer solutions (n , m) with n ≥ 0 and m ≥ 0 , except for the only pair (c , d) = (− 1 , 2). Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai's problem in linear recurrent sequences. [ABSTRACT FROM AUTHOR]

We present here a general rule of construction of identities for recursive sequences by using sequence transformation techniques developed in [16]. Numerous identities are constructed, and many well known identities can be proved readily by using this unified rule. Various Catalan-like and Cassini-like identities are given for recursive number sequences and recursive polynomial sequences. Sets of identities for Diophantine quadruple are shown. [ABSTRACT FROM AUTHOR]

It is well known that one of the most relevant applications of the reverse Bessel polynomials θn(z) is filter design. In particular, the poles of the transfer function of a Bessel filter are basically the zeros of θn(z). In this article we discuss an algorithm to compute the zeros of reverse generalized Bessel polynomials θn(z;a). A key ingredient in the algorithm will be a method to compute the polynomials. For this purpose, we analyze the use of recurrence relations and asymptotic expansions in terms of elementary functions to obtain accurate approximations to the polynomials. The performance of all the numerical approximations will be illustrated with examples. [ABSTRACT FROM AUTHOR]