Related papers: Cassini, d'Ocagne, and Vajda identities for n-step…
In this paper, we show that the classical Cassini and Catalan identities for Fibonacci numbers arise naturally from a single quadratic theta-function identity of Ramanujan. Expanding the identity…
We extend our investigation of $2$-determinants, which we defined in a previous paper. For a linear homogenous recurrence of the second order, we consider relations between different sequences satisfying the same linear homogeneous…
We evaluate a determinant of generalized Fibonacci numbers, thus providing a common generalization of several determinant evaluation results that have previously appeared in the literature, all of them extending Cassini's identity for…
We apply the Desnanot-Jacobi identity to give an alternative proof of the determinants whose entries are rising powers of the Fibonacci numbers given by Prodinger. We then generalize the determinants to include entries that are rising…
We present a differential-calculus-based method which allows one to derive more identities from {\it any} given Fibonacci-Lucas identity containing a finite number of terms and having at least one free index. The method has two {\it…
In this paper, hyperbolic k-Fibonacci quaternions are defined. Also, some algebraic properties of hyperbolic k-Fibonacci quaternions which are connected with hyperbolic numbers and k-Fibonacci numbers are investigated. Furthermore,…
In this paper, bicomplex k-Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex k-Fibonacci quaternions which are connected with bicomplex numbers and k-Fibonacci numbers are investigated. Furthermore, the…
We derive identities for the determinants of matrices whose entries are (rising) powers of (products of) polynomials that satisfy a recurrence relation. In particular, these results cover the cases for Fibonacci polynomials, Lucas…
We derive a new Fibonacci identity. This single identity subsumes important known identities such as those of Catalan, Ruggles, Halton and others, as well as standard general identities found in the books by Vajda, Koshy and others. We also…
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
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…
In this paper, we evaluate determinants of some families of Toeplitz-Hessenberg matrices having tribonacci number entries. These determinant formulas may also be expressed equivalently as identities that involve sums of products of…
We give a new Jacobi--Trudi-type formula for characters of finite-dimensional irreducible representations in type $C_n$ using characters of the fundamental representations and non-intersecting lattice paths. We give equivalent determinant…
In this paper, we find several determinants expressing the Fibonomial coefficients. We also give the generating functions, Vandermonde identity, and continued fractions about Fibonomial coefficients.
This (partly expository) paper originated from the study of Hankel determinants of convolution powers of Catalan numbers and of Narayana polynomials. This led to some Hankel determinants of signed Catalan numbers whose values are multiples…
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 paper we evaluate some Toeplitz-type determinants. Let $n>1$ be an integer. We prove the following two basic identities: \begin{align*} \det{[j-k+\delta_{jk}]_{1\leq j,k\leq n}}&=1+\frac{n^2(n^2-1)}{12}, \\…
The Jacobi identities play an important role in constructing the explicit exact solutions of a broad class of integrable systems in soliton theory. In the paper, a direct and simple proof of the Jacobi identities for determinants is…
We extend Fibonacci numbers with arbitrary weights and generalize a dozen Fibonacci identities. As a special case, we propose an elliptic extension which extends the $q$-Fibonacci polynomials appearing in Schur's work. The proofs of most of…
The Fibonacci polynomials are defined recursively as $f_{n}(x)=xf_{n-1}(x)+f_{n-2}(x)$, where $f_0(x) = 0$ and $f_1(x)= 1$. We generalize these polynomials to an arbitrary number of variables with the $r$-Fibonacci polynomial. We extend…