On generalized Fibonacci and Lucas polynomials

Let h(x) be a polynomial with real coefficients. We introduce h(x)-Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h(x)-Fibonacci polynomials. We also introduce h(x)-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q_h(x) that generalizes the Q-matrix whose powers generate the Fibonacci numbers.     

A topological approach to divisibility of arithmetical functions and GCD matrices

Considering lower closed sets as closed sets on a preposet (P, ≤), we obtain an Alexandroff topology on P. Then order preserving functions are continuous functions. In this article we investigate order preserving properties (and thus continuity properties) of integer-valued arithmetical functions under the usual divisibility relation of integers and power GCD matrices under the divisibility relation of integer matrices.     

On unitary analogs of GCD reciprocal LCM matrices

A divisor d ∈ ?₊ of n ∈ ?₊ is said to be a unitary divisor of n if (d, n/d) = 1. In this article we examine the greatest common unitary divisor (GCUD) reciprocal least common unitary multiple (LCUM) matrices. At first we concentrate on the difficulty of the non-existence of the LCUM and we present three different ways to overcome this difficulty. After that we calculate the determinant of the three GCUD reciprocal LCUM matrices with respect to certain types of functions arising from the LCUM problematics. We also analyse these classes of functions, which may be referred to as unitary analogs of the class of semimultiplicative functions, and find their connections to rational arithmetical functions. Our study shows that it does make a difference how to extend the concept of LCUM.     

An analogue of Ramanujan’s sum with respect to regular integers (modr)

An integer a is said to be regular (modr) if there exists an integer x such that a ² x≡a (mod r). In this paper we introduce an analogue of Ramanujan’s sum with respect to regular integers (modr) and show that this analogue possesses properties similar to those of the usual Ramanujan’s sum.     

An arithmetical equation with respect to regular convolutions

It is well known that Euler’s totient function \(\phi \) satisfies the arithmetical equation \( \phi (mn)\phi ((m, n))=\phi (m)\phi (n)(m, n) \) for all positive integers m and n, where (m, n) denotes the greatest common divisor of m and n. In this paper we consider this equation in a more general setting by characterizing the arithmetical functions f with \(f(1)\ne 0\) which satisfy the arithmetical equation \( f(mn)f((m,n)) = f(m)f(n)g((m, n)) \) for all positive integers m, n with \(m,n \in A(mn)\), where A is a regular convolution and g is an A-multiplicative function. Euler’s totient function \(\phi _A\) with respect to A is an example satisfying this equation.     

Some analogues of smith's determinant

We calculate the determinants of the greatest common divisor (GCD) and the least common multiple (LCM) matrices associated with an arithmetical function on gcd-closed and lcm-closed sets. We also consider some unitary analogues of these determinants.     

A generalization of Descartes’ rule of signs and fundamental theorem of algebra

Descartes’ rule of signs yields an upper bound for the number of positive and negative real roots of a given polynomial. The fundamental theorem of algebra implies a similar property; every real polynomial of degree n ? 1 has at most n real zeroes. In this paper, we describe axiomatically function families possessing one or another of these properties. The resulting families include, at least, all polynomial functions and sums of exponential functions. As an application of our approach, we consider, among other things, a method for identifying certain type of bases for the Euclidean space.