The main aim of this paper is to introduce generalized quaternions whose components are higher-order generalized Fibonacci hyper-numbers with -integer numbers. We first derive the Binet-like formula for this newly established type of generalized quaternions. With the help of the Binet-like formula, we give recurrence relation, generating function, exponential generating function, and some useful summation properties of these generalized quaternions.

The aim of this paper is to introduce Leonardo finite operator polynomials and obtain some of their new properties. We first present the recurrence relation provided by Leonardo finite operator polynomials. Then, we give a Binet-like formula, generating function, exponential generating function, and a finite sum formula for Leonardo finite operator polynomials. We present a determinant representation for the nth term of Leonardo finite operator polynomials. Ultimately, by utilizing the generating function of the proposed polynomials, we establish generating relations for specific bilinear and bilateral polynomial families. This approach thus broadens the applicability of the finite operator framework to encompass a wider range of special functions.

Many properties of special numbers, such as sum formulas, symmetric properties, and their relationships with each other, have been studied in the literature with the help of the Binet formula and generating function. In this paper, higher-order generalized Fibonacci hybrid numbers with q-integer components are defined through the utilization of q-integers and higher-order generalized Fibonacci numbers. Several special cases of these newly established hybrid numbers are presented. The article explores the integration of q-calculus and hybrid numbers, resulting in the derivation of a Binet-like formula, novel identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of hybrid numbers with quantum integer coefficients. Furthermore, new identities for these types of hybrids are obtained using two novel special matrices. To substantiate the findings, numerical examples are provided, generated with the assistance of Maple.

In this paper, we introduce hybrid numbers with Fibonacci and Lucas hybrid number coefficients. We give the Binet formulas, generating functions, exponential generating functions for these numbers. Then we define an associate matrix for these numbers. In addition, using this matrix, we present two different versions of Cassini identitiy of these numbers.

In this paper, we investigate a min matrix and obtain its $ LU $-decomposition, determinant, permanent, inverse, and norm properties. In addition, we obtain a recurrence relation provided by the characteristic polynomial of this matrix. Finally, we present an example to illustrate the results obtained.

A number of families of quaternion and octonion number sequence such as (Fibonacci quaternion, Fibonacci octonion and so forth) have been studied by several authors in many different ways. Besides, several formulas and identities involving these number sequences have been presented. The aim of this paper is to consider the octonions with components including quantum integers. We called these type of octonions the q-Fibonacci octonions and the q-Lucas octonions respectively. Furthermore, we give Binet formulas, exponential generating functions, summation formulas, Catalan identities, Cassini identities and d'Ocagne identities, respectively.

Irmak recently asked an open question related to divisibility properties of Fibonacci and Lucas quaternions [4, p. 374]. In this paper, we give an answer to Fibonacci and Lucas hybrid number version of this question.

In the paper, the authors find closed formulas and recurrent relations for bi-periodic Fibonacci polynomials and for bi-periodic Lucas polynomials in terms of the Hessenberg determinants. Consequently, the authors derive closed formulas and recurrent relations for the Fibonacci, Lucas, bi-periodic Fibonacci, and bi-periodic Lucas numbers in terms of the Hessenberg determinants.

In this study, we obtain upper and lower bounds for the spectral norms of the geometric circulant matrices with the bi--periodic Fibonacci numbers and bi--periodic Lucas numbers, respectively. Then we give some bounds for the spectral norms of Kronecker and Hadamard products of these matrices.

In this paper, we give lower and upper bounds for the spectral norms of the r-circulant matrices whose entries are B-n,B-k = k/n((2n)(n - k)) and kB(n,k) = k(2)/n((2n)(n - k)), respectively, where n is an element of N, k = 0, 1, 2, . . . , n-1, k <= n. Then we present some bounds for the spectral norms of Kronecker and Hadamard products of these matrices.

In this paper, we give a generalization of the Fibonacci and Lucas quaternions. We obtain the Binet formulas, generating functions, and some certain identities for these quaternions which include generalizations of some results of Halici.

In this paper, we introduce the split k-Fibonacci and k-Lucas quaternions. We obtain the Binet formulas, generating functions and exponential generating functions of these quaternions. Moreover, we give the Catalan, Cassini and d'Ocagne identities for the split k-Fibonacci and k-Lucas quaternions.

In this paper, we give the exponential generating functions for the generalized Fibonacci and generalized Lucas quaternions, respectively. Moreover, we give some new formulas for binomial sums of these quaternions by using their Binet forms.

Ramirez recently conjectured a version of Catalan's identity for the k-Fibonacci quaternions. In this note we give a proof of (a suitably reformulated version of) this identity.