[n is an arbitrary natural number, r is an arbitrary (nonzero) real quantity) gives & positive integer k. Since both r and k turn out to be Fibonacci number ratios, the results established in this paper can be viewed as a particular kind of Fibonacci identities that are believed to be new [see (4.7) and (4.8)]. Throughout the paper we shall make use of the following properties of the Fibonacci ...

In this paper we present some results on the annular bound for the zeros of a polynomial based on the identities related to the generalized Fibonacci sequence with arbitrary initial condition. Several recently reported results in the same direction are special cases of our results.

In this paper, we establish a formula expressing explicitly the general term of a linear recurrent sequence, allowing us to generalize the original result of J. McLaughlin [7] concerning powers of a matrix of size 2, to the case of a square matrix of size m ≥ 2. Identities concerning Fibonacci and Stirling numbers and various combinatorial relations are derived.

In this paper, with the help of finite operators and Fibonacci numbers, we define a new family quaternions whose components are operator numbers. We also provide some properties these types quaternions. Moreover, derive many identities related to by using matrix representations.

Abstract. MacMahon’s definition of self-inverse composition is extended to n-colour self-inverse composition. This introduces four new sequences which satisfy the same recurrence relation with different initial conditions like the famous Fibonacci and Lucas sequences. For these new sequences explicit formulas, recurrence relations, generating functions and a summation formula are obtained. Two ...

The aim of this paper is to present a comprehensive survey cubic Fibonacci identities, trying uncover as many possible. From the outset, our rationale for very careful search on an apparently obscure problem was not only matter mathematical curiosity, but also motivated by quest 3D spirals.
 As we were able find any particular topic identities decided try fill void. We started surveying an...

In this paper, we will derive the explicit formulae for Chebyshev polynomials of third and fourth kind with odd even indices using combinatorial method. Similar results are also deduced their r-th derivatives. Finally, some identities involving Fibonacci negative obtained.