The Hadamard Products for bi-periodic Fibonacci and bi-periodic Lucas Generating matrices

In this paper, firstly, we define the Qq-generating matrix for bi-periodic Fibonacci polynomial. And we give nth power, determinant and some properties of the bi-periodic Fibonacci polynomial by considering this matrix representation. Also, we introduce the Hadamard products for bi-periodic Fibonacci Qnq generating matrix and bi-periodic Lucas Qnl generating matrix of which entries is bi-periodic Fibonacci and Lucas numbers. Then, we investigate some properties of these products

New hypergeometric connection formulae between Fibonacci and Chebyshev polynomials

We establish new connection formulae between Fibonacci polynomials and Chebyshev polynomials of the first and second kinds. These formulae are expressed in terms of certain values of hypergeometric functions of the type 2F1. Consequently, we obtain some new expressions for the celebrated Fibonacci numbers and their derivatives sequences. Moreover, we evaluate some definite integrals involving products of Fibonacci and Chebyshev polynomials

On the binomial sums of Horadam sequence

The main purpose of this paper is to establish some new properties of Horadam numbers in terms of binomial sums. By that, we can obtain these special numbers in a new and direct way. Moreover, some connections between Horadam and generalized Lucas numbers are revealed to get a more strong result

Extremal orders of the Zeckendorf sum of digits of powers

Denote by s_F(n) the minimal number of Fibonacci numbers needed to write n as a sum of Fibonacci numbers. We obtain the extremal minimal and maximal orders of magnitude of s_F(n^h)/s_F(n) for any h>= 2. We use this to show that for all $>N_0(h) there is an n such that n is the sum of N Fibonacci numbers and n^h is the sum of at most 130 h^2 Fibonacci numbers. Moreover, we give upper and lower bounds on the number of n's with small and large values of s_F(n^h)/s_F(n). This extends a problem of Stolarsky to the Zeckendorf representation of powers, and it is in line with the classical investigation of finding perfect powers among the Fibonacci numbers and their finite sums.Comment: 11 page

Star of David and other patterns in the Hosoya-like polynomials triangles

In this paper we first generalize the numerical recurrence relation given by Hosoya to polynomials. Using this generalization we construct a Hosoya-like triangle for polynomials, where its entries are products of generalized Fibonacci polynomials (GFP). Examples of GFP are: Fibonacci polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell polynomials, Fermat polynomials, Jacobsthal polynomials, Vieta polynomials and other familiar sequences of polynomials. For every choice of a GFP we obtain a triangular array of polynomials. In this paper we extend the star of David property, also called the Hoggatt-Hansell identity, to this type of triangles. We also establish the star of David property in the gibonomial triangle. In addition, we study other geometric patterns in these triangles and as a consequence we give geometric interpretations for the Cassini's identity, Catalan's identity, and other identities for Fibonacci polynomials.Comment: Eight Figures and 24 page

Some Properties of Horadam quaternions

In this paper, we consider the generalized Fibonacci quaternion which is the Horadam quaternion sequence. Then we used the Binet's formula to show some properties of the Horadam quaternion. We get some generalized identities of the Horadam number and generalized Fibonacci quuaternion

On Dirichlet Products Evaluated at Fibonacci Numbers

In this work we discuss Dirichlet products evaluated at Fibonacci numbers. As first applications of the results we get a representation of Fibonacci numbers in terms of Euler's totient function, an upper bound on the number of primitive prime divisors and representations of some related Euler products. Moreover, we sum functions over all primitive divisors of a Fibonacci number and obtain a non--trivial fixed point of this operation.Comment: 22 pages, 1 figur

Repdigits as products of consecutive balancing or Lucas-balancing numbers

Repdigits are natural numbers formed by the repetition of a single digit. In this paper, we explore the presence of repdigits in the product of consecutive balancing or Lucas-balancing numbers

Chords of an ellipse, Lucas polynomials, and cubic equations

A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse, generalizing a classic problem about circles. We give a brief history of the circle problem, an account of Price's ellipse proof, and a reorganized proof, with some new ideas, designed to situate the result within a dense web of connections to classical mathematics. It is inspired by Cardano's solution of the cubic equation and Newton's theorem on power sums, and yields an interpretation of generalized Lucas polynomials in terms of the theory of symmetric polynomials. We also develop additional connections that surface along the way; e.g., we give a parallel interpretation of generalized Fibonacci polynomials, and we show that Cardano's method can be used write down the roots of the Lucas polynomials.Comment: 16 pages, 1 table, 2 figures. A substantially revised version of this paper has now been accepted for publication in the American Mathematical Monthly. Per Taylor and Francis policy, we are unable to post the accepted version here until 12 months after publication. We will add the journal reference and DOI when the article is publishe

Sums of products of generalized Fibonacci and Lucas numbers

In this paper, we establish several formulae for sums and alternating sums of products of generalized Fibonacci and Lucas numbers. In particular, we recover and extend all results of Z. Cerin and Z. Cerin & G. M. Gianella, more easily