Related papers: Differentiating a Linear Recursive Sequence
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…
Causal functions of sequences occur throughout computer science, from theory to hardware to machine learning. Mealy machines, synchronous digital circuits, signal flow graphs, and recurrent neural networks all have behaviour that can be…
Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function.…
A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences…
In this study, depending on the upper and the lower indices of the hyperharmonic number $h_{n}^{(r)}$, nonlinear recurrence relations are obtained. It is shown that generalized harmonic number and hyperharmonic number can be obtained from…
High-order derivatives of nested functions of a single variable can be computed with the celebrated Fa\`a di Bruno's formula. Although generalizations of such formula to multiple variables exist, their combinatorial nature generates an…
In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured…
Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence…
This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular…
The results presented in this paper are refinements of some results presented in a previous paper. Three such refined results are presented. The first one relaxes one of the basic hypotheses assumed in the previous paper, and thus extends…
We prove that the uniform recurrence of morphic sequences is decidable. For this we show that the number of derived sequences of uniformly recurrent morphic sequences is bounded. As a corollary we obtain that uniformly recurrent morphic…
Second order recurrence relations of real numbers arise form various applications in discrete time dynamical systems as well as in the context on Markov chains. Solutions to the recurrence relations are fully defined by the first two…
New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high…
Primitive recursion is a mature, well-understood topic in the theory and practice of programming. Yet its dual, primitive corecursion, is underappreciated and still seen as exotic. We aim to put them both on equal footing by giving a…
We study nonlinear regression of real valued data in an individual sequence manner, where we provide results that are guaranteed to hold without any statistical assumptions. We address the convergence and undertraining issues of…
We present in this paper a detailed note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation. This paper is a continuation of [1] which was on the…
Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left…
We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between…
Many modern numerical methods in computational science and engineering rely on derivatives of mathematical models for the phenomena under investigation. The computation of these derivatives often represents the bottleneck in terms of…
We study linear difference equations with variable coefficients in a ring using a new nonlinear method. In a ring with identity, if the homogeneous part of the linear equation has a solution in the unit group of the ring (i.e., a unitary…