Related papers: $n$-Colour self-inverse compositions
We develop new closed form representations of sums of (n + {\alpha})th shifted harmonic numbers and reciprocal binomial coefficients in terms of {\alpha}th shifted harmonic numbers. Some interesting new consequences and illustrative…
Nicholas Pippenger and Kristin Schleich have recently given a combinatorial interpretation for the second-order super-Catalan numbers (u_{n})_{n>=0}=(3,2,3,6,14,36,...): they count "aligned cubic trees" on n internal vertices. Here we give…
In this paper, we define the incomplete h(x)-Fibonacci and h(x)-Lucas polynomials, we study recurrence relations and some properties of these polynomials
We define a family of meta-Fibonacci sequences where the order of the of recursion at stage n is a variable r(n), and the n^{th} term of a sequence is the sum of the previous r(n) terms. For the terms of any such sequence, we give upper and…
A class of determinants is introduced. Different kind of mathematical objects, such as Fibonacci, Lucas, Tchebychev, Hermite, Laguerre, Legendre polynomials, sums and covergents are represented as determinants from this class. A closed…
The purpose of this paper is twofold. First, the definition of new statistical convergence with Fibonacci sequence is given and some fundamental properties of statistical convergence are examined. Second, approximation theory worked as a…
Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials…
A natural class of coloring complexes $X$ on closed manifold $M^n$ is investigated that gives a holonomy map $\mbox{Hol}_X: \pi_1(M) \to S_{n+1}$. By a $k$-multilayer complex construction the holonomy map may be defined to any finite…
Studying and comparing arithmetic properties of a given automatic sequence and the sequence of coefficients of the composition inverse of the associated formal power series (the formal inverse of that sequence) is an interesting problem.…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting…
In the same way decomposition spaces, also known as unital 2-Segal spaces, have incidence (co)algebras, and certain relative decomposition spaces have incidence (co)modules, we identify the structures that have incidence bi(co)modules: they…
In this paper we introduce mixed coloured permutation, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and…
A recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed. We show that infinitely many recurrence equations can be derived from the information about invariant varieties of periodic…
We study compositional inverses of permutation polynomials, complete mappings, mutually orthogonal Latin squares, and bent vectorial functions. Recently it was obtained in [33] the compositional inverses of linearized permutation binomials…
We define iteration of functions that map n-dimensional vector spaces into m-dimensional vector spaces (m at most equal to n). It happens that usual iteration and Fibonacci iterative methods become special cases of this generalized…
Bayes' rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may ask whether composing the inversions of the component…
The aim of this article is to investigate the issues of multiplicative inverses and composition in the set of formal Laurent series. We show the lack of general uniqueness of inverses of formal Laurent series; necessary and sufficient…
For the Lucas sequence $\{U_{k}(P,Q)\}$ we discuss the identities such as the well-known Fibonacci identities. We also propose a method for obtaining identities involving recurrence sequences. With the help of which we find an interpolating…
By studying Cameron's operator in terms of determinants, two kinds of "integer" sequences of incomplete numbers were introduced. One was the sequence of restricted numbers, including $s$-step Fibonacci sequences. Another was the sequence of…