Related papers: On a four-parameters generalization of some specia…
By investigating a recurrence relation about functions, we first give alternative proofs of various identities on Fibonacci numbers and Lucas numbers, and then, make certain well known identities visible via certain trivalent graph…
The Fibonacci sequence is obtained as weighted sum along the rows in the Pascal triangle by choosing a periodic up-and-down pattern of weights from the set $\{-1,-\frac{1}{2},0, \frac{1}{2}, 1\}$. A graphical illustration of this identity…
Since the $\mathrm{Fibonacci}$ sequence has good properties, it's important in theory and applications, such as in combinatorics, cryptography, and so on. In this paper, for the generalized Fibonacci sequence…
The Lucas polynomials, $\{n\}$, are polynomials in $s$ and $t$ given by $\{ n \} = s \{ n-1 \} + t \{ n-2 \}$ for $n \geq 2$ with $ \{ 0 \} = 0$ and $\{ 1 \} = 1$. The lucanomial coefficients, an analogue of the binomial coefficients, are…
We present a parametric family of Riordan arrays which are obtained by multiplying any Riordan array with a generalized Pascal array. In particular, we focus on some interesting properties of one-parameter Catalan triangles. We obtain…
We study matrices which transform the sequence of Fibonacci or Lucas polynomials with even index to those with odd index and vice versa. They turn out to be intimately related to generalized Stirling numbers and to Bernoulli, Genocchi and…
Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of…
In this paper we determine some properties of Fibonacci octonions. Also, we introduce the generalized Fibonacci-Lucas octonions and we investigate some properties of these elements.
A new family of generalized Pell numbers was recently introduced and studied by Br\'od \cite{Dorota}. These number possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can…
We investigate general properties of number sequences which allow explicit representation in terms of products. We find that such sequences form whole families of number sequences sharing similar recursive identities. Restricting to the…
We describe how to compute the intersection of two Lucas sequences of the forms $\{U_n(P,\pm 1) \}_{n=0}^{\infty}$ or $\{V_n(P,\pm 1) \}_{n=0}^{\infty}$ with $P\in\mathbb{Z}$ that includes sequences of Fibonacci, Pell, Lucas, and Lucas-Pell…
In this paper, we introduce a novel identity for generalized Euler polynomials, leading to further generalizations for several relations involving classical Euler numbers, Euler polynomials, Genocchi polynomials, and Genocchi numbers.
In part 1 of this paper some linear weighted generalized Fibonacci number summation identities were derived using the fact that the Fibonacci number is the residue of a rational function. In this part, using the same method, some quadratic…
One of the best known and oldest identities for the Fibonacci sequence $F_n$ is $F_{n+1}F_{n-1}-F_{n}^2=(-1)^n$ which was derived first by R. Simson in 1753 and it is now called as Simson or Cassini Identity. In this paper, we generalize…
In this study, we introduce the generalized Gaussian third-order Jacobsthal numbers with arbitrary initial values and discuss two particular cases, namely, Gaussian third-order Jacobsthal and Gaussian modified third-order Jacobsthal…
We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in…
In this study, we introduce the generalized Tribonacci hyperbolic spinors and properties of this new special numbers system by the generalized Tribonacci numbers, which are one of the most general form of the third-order recurrence…
We derive some Fibonacci and Lucas identities which contain inverse binomial coefficients. Extension of the results to the general Horadam sequence is possible, in some cases.
In this paper, we establish an identity for Bernoulli's generalized polynomials. We deduce generalizations for many relations involving classical Bernoulli numbers or polynomials. In particular, we generalize a recent Gessel identity.
In this paper we extend a notion of Cassini determinant to recently introduced hyperfibonacci sequences. We find $Q$-matrix for the $r$-th generation hyperfibonacci numbers and prove an explicit expression of the Cassini determinant for…