Related papers: On the geometry behind a recurrent relation
This paper proves that two differently defined rooted binary trees are isomorphic. The first tree is one associated to a version of Farey sequences where the vertices correspond to the open intervals formed by two successive terms in the…
We discuss some special property of the Farey sequence. We show in each term of the Farey sequence, ratio of the sum of elements in the denominator and the sum of elements in the numerator is exactly two. We also show that the Farey…
This paper introduces Farey Recursive Functions and investigates their basic properties. Farey Recursive Functions are a special type of recursive function from the rationals to a commutative ring. The recursion of these functions is…
Six families of generalized hypergeometric series in a variable $x$ and an arbitrary number of parameters are considered. Each of them is indexed by an integer $n$. Linear recurrence relations in $n$ relate these functions and their product…
The Farey sequence is a natural exhaustion of the set of rational numbers between 0 and 1 by finite lists. Ford Circles are a natural family of mutually tangent circles associated to Farey fractions: they are an important object of study in…
In this paper, we give recurrence relations and identities for some integer sequences related to Ward numbers such as Ward-Lah numbers, varied Ward numbers and binomial Ward numbers. Most of the sequences are entered in the On-Line…
We define recurrence matrices and study a few properties (links with automatic sequences, branch groups etc.) of them.
In this note we study a family of algebras with one parameter defined by generators and relations. The set of generators contains the generators of the usual braids algebra, and another set of generators which is interpreted as ties between…
We present a certain generalization of a recent result of M. I. Cirnu on linear recurrence relations with coefficient in progressions [2]. We provide some interesting examples related to some well-known integer sequences, such as Fibonacci…
When a series of measurements is performed with increasingly coarse (or increasingly fine) precision, consecutive observations seem to be erratically distributed at first, and then organize themselves into cycles and patterns. The patterns,…
We present further properties of a previously proposed recursive scheme for parameterisation of n-by-n unitary matrices. We show that the factors in the recursive formula may be introduced in any desired order. The method is used to study…
For a hypergeometric series $\sum_k f(k,a, b, ...,c)$ with parameters $a, b, >...,c$, Paule has found a variation of Zeilberger's algorithm to establish recurrence relations involving shifts on the parameters. We consider a more general…
The 15 Gauss contiguous relations for ${}_2F_1$ hypergeometric series imply that any three ${}_2F_1$ series whose corresponding parameters differ by integers are linearly related (over the field of rational functions in the parameters). We…
Recently it has been found that some special subsequences within a Farey sequence play a crucial role in determining the ranges of coupling constant for which quantum soliton states can exist for an integrable derivative nonlinear…
Through the following, we establish the conditions which allow us to express recursive sequences of real numbers, enumerated through the recurrence relation a_{n+1} = Aa_n + Ba_{n-1}, by means of algebraic equations in two variables of…
We introduce a method for describing Riordan matrices via recurrence relations along their diagonals. This provides a new structural description that complements the classical row-wise and column-wise constructions via the A-sequence. As an…
We present a finite-order system of recurrence relations for a permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k = 1, 2, and 3) as well as the method for deriving such recurrence…
The problem of calculating multicanonical parameters recursively is discussed. I describe in detail a computational implementation which has worked reasonably well in practice.
A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. We consider a family of nonlinear recurrences with the Laurent property, which were…
Number sequences defined by a linear recursion relation are studied by means of generating functions. Indices of the terms in the recursion relation have arbitrary differenses. In addition to formulas for the nth term an algorithm is…