We consider the power sums, alternating power sums, and product power sums of the bi-periodic Fibonacci and Lucas polynomials by Sury's polynomial identity. Furthermore, we obtain the divisibility properties for the bi-periodic Fibonacci and Lucas polynomials.
In 2012, Kuhapatanakul established striking double-sum identities for the square and the product of two consecutive generalized Fibonacci numbers. His elegant, purely combinatorial proofs relied on properties of the generalized Tribonacci triangle. In this note, we provide concise and self-contained proofs of these identities using generating functions, and discuss several natural extensions and related formulas.
Fibonacci and Lucas numbers have been extensively studied in various algebraic and number-theoretic contexts, including their modular inverses and generalizations. Motivated by these developments, we study the inverses of Fibonacci and Lucas numbers with rational indices through the codenominator function F. Uludağ and Gökmen (2022) showed that the rational-indexed Fibonacci number F_X, where X∈ℚ_>0, can be expressed using F, and that infinitely many such representations exist. In this paper, we extend their work by deriving a general explicit formula for F_X through the codenominator function. We establish precise conditions under which F_X coincides with a Lucas number or deviates from the classical Fibonacci sequence. Moreover, by means of these generalized formulas, we compute the units of Fibonacci and Lucas numbers, thereby proving the existence of multiplicative inverses for all Fibonacci and Lucas numbers within this framework. These results reveal new structural properties of Fibonacci- and Lucas-related sequences and suggest further directions for number-theoretic exploration.
We prove many identities involving sums with products of Gibonacci numbers in the denominator. Three of our results provide generalizations of problems published in The Fibonacci Quarterly. We also study Brousseau sums with Gibonacci entries.
In 1957 George Bergman demonstrated that every positive integer has a terminating representation in base phi = (1 + root 5) /2, with each digit d(i) being either 0 or 1 and no two consecutive digits both being 1. This paper expands this result by establishing the existence and properties of representations of all positive real numbers in base phi and in other related irrational bases.
Motivated by a recent problem in the American Mathematical Monthly, we find simple formulas for the sums of several large collections of infinite series involving the arctangent of the Fibonacci and Lucas numbers.
This article serves as a companion to the video essay [9] by Sheafification of G on computing large Fibonacci numbers quickly. We follow the same progression of methods-from naive recursion to sophisticated Fourier transforms-and elaborate the concepts that were impractical to discuss in video format.
We prove a conjecture due to Gica [2]. Our proof is simple, uses only elementary abstract algebra, and generalizes to other recursive sequences.
