We extend Fibonacci numbers with arbitrary weights and generalize a dozen identities. As special case, we propose an elliptic extension which extends the q-Fibonacci polynomials appearing in Schur's work. The proofs of most identities are combinatorial, extending given by Benjamin Quinn, q-case, Garrett. Some proved telescoping.

Hypergeometric functions are an important tool in many branches of pure and applied mathematics, and they encompass most special functions, including the Chebyshev polynomials. There are also well-known connections between Chebyshev polynomials and sequences of numbers and polynomials related to Fibonacci numbers. However, to my knowledge and with one small exception, direct connections between...

In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving binomial coefficients and Fibonacci type sequences.

We formalized some basic properties of the Fibonacci numbers using definitions and lemmas from [7] and [23], e.g. Cassini’s and Catalan’s identities. We also showed the connections between Fibonacci numbers and Pythagorean triples as defined in [31]. The main result of this article is a proof of Carmichael’s Theorem on prime divisors of prime-generated Fibonacci numbers. According to it, if we ...

In 1982, Prodinger and Tichy defined the Fibonacci number of a graph G to be the number of independent sets of the graph G. They did so since the Fibonacci number of the path graph Pn is the Fibonacci number Fn+2 and the Fibonacci number of the cycle graph Cn is the Lucas number Ln. The tadpole graph Tn,k is the graph created by concatenating Cn and Pk with an edge from any vertex of Cn to a pe...