相关论文: Some power series extensions of Fibonacci and Luca…
We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.
Powers of Fibonacci polynomials are expressed as single sums, improving on a double sum recently seen in the literature.
We show that certain weighted Fibonacci and Lucas series can always be expressed as linear combinations of polylogarithms. In some special cases we evaluate the series in terms of Bernoulli polynomials, making use of the connection between…
We study certain series with Catalan numbers and reciprocal Catalan numbers, respectively, and provide seemingly new closed form evaluations of these series with Fibonacci (Lucas) entries. In addition, we state some combinatorial sums that…
This note gives an elementary exposition of a variant of the spread polynomials in terms of Fibonacci and Lucas polynomials.
We study sums of powers of Fibonacci and Lucas polynomials of the form $% \sum_{n=0}^{q}F_{tsn}^{k}(x) $ and $\sum_{n=0}^{q}L_{tsn}^{k}% (x) $, where $s,t,k$ are given natural numbers, together with the corresponding alternating sums…
Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.
We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for $\pi$ involving Fibonacci and Lucas numbers. Our findings complement those…
We give an overview about well-known basic properties of two classes of q-Fibonacci and q-Lucas polynomials and offer a common generalization.
We study the interpolation group whose elements are suitable pairs of formal power series. This group has a faithful representation into infinite lower triangular matrices and carries thus a natural structure as a Lie group. The matrix…
We define the convolved h(x)-Fibonacci polynomials as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the h(x)-Fibonacci and h(x)-Lucas polynomials. Moreover we obtain the…
This paper presents new identities expressing the terms of Fibonacci, Lucas, and generalized Fibonacci sequences with multiple indices through powers of Lucas numbers and binomial coefficients. The obtained formulas rely on the application…
In this paper, we provide new applications of Fibonacci and Lucas numbers. In some circumstances, we find algebraic structures on some sets defined with these numbers, we generalize Fibonacci and Lucas numbers by using an arbitrary binary…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
Bugeaud, Mignotte, and Siksek proved that the only perfect powers in Fibonacci sequence are 0, 1, 8, and 144. In this paper, we study the polynomial analogue of the problem. Especially, we give a complete characterization of the Fibonacci…
We consider $m$-th order linear recurrences that can be thought of as generalizations of the Lucas sequence. We exploit some interplay with matrices that again can be considered generalizations of the Fibonacci matrix. We introduce the…
A simple matrix formulation of the Fibonacci, Lucas, Chebyshev, and Dixon polynomials polynomials is presented. It utilizes the powers and the symmetric tensor powers of a certain matrix.
The concept of a fully interlacing matrix of formal power series with real coefficients is introduced. This concept extends and strengthens that of an interlacing sequence of real-rooted polynomials with nonnegative coefficients, in the…
In this study, we define a new type of Fibonacci and Lucas num- bers which are called bicomplex Fibonacci and bicomplex Lucas numbers. We obtain the well-known properties e.g. Docagnes, Cassini, Catalan for these new types. We also give the…
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.