Emine Gökçen KOÇER

• Necmettin Erbakan University

The group of the almost-Riordan arrays with exponential generating functions is defined. The subgroups of the exponential almost-Riordan group are presented. Also, some isomorphisms between the exponential almost-Riordan group and the exponential Riordan group are considered. Then, the production matrix for the exponential almost-Riordan array is o...

UDC 511 A generalization of the Leonardo numbers is defined and called the hyper-Leonardo numbers. Infinite lower triangular matrices, whose elements are Leonardo and hyper-Leonardo numbers are considered. Then the A - and Z -sequences of these matrices are obtained. Finally, the combinatorial identities between the hyper-Leonardo and Fibonacci num...

In this study, we consider the Hankel and Gram matrices which are defined by the elements of special number sequences. Firstly, the eigenvalues, determinant, and norms of the Hankel matrix defined by special number sequences are obtained. Afterwards, using the relationship between the Gram and Hankel matrices, the eigenvalues, determinants, and nor...

In this paper, we define a generalization of the Leonardo numbers called the hyper-Leonardo numbers. Then, the infinite lower triangular matrices with Leonardo and hyper-Leonardo numbers defined and found the A and Z−sequences of these matrices. Moreover, by using the fundamental theorem of Riordan arrays, some combinatorial identities obtained. Fi...

Bivariate Leonardo polynomials, which are closely related to bivariate Fibonacci polynomials, are defined in this paper. Bivariate Leonardo polynomials are a generalization of the Leonardo polynomials and Leonardo numbers. We obtained some properties and identities (Cassini, Catalan, Honsberger, d'Ocagne) for the bivariate Leonardo polynomials.Then...

The hybrid numbers are a generalization of complex, hyperbolic and dual numbers. Until this time, many researchers have studied related to hybrid numbers. In this paper, using the generalized Fibonacci and Lucas p-numbers, we introduce the generalized hybrid Fibonacci and Lucas p-numbers. Also, we give some special cases and algebraic properties of...

Until today, many researchers have studied related to hybrid numbers which are a generalization of complex, hyperbolic and dual numbers. In this paper, using the Leonardo numbers, we introduce the hybrid Leonardo numbers. Also, we give some algebraic properties of the hybrid Leonardo numbers such as recurrence relation, generating function, Binet’s...

In this paper, we consider the Leonardo numbers which is defined by Catarino and Borges. Using Binet's formula of this sequence, we obtain new identities of the Leonardo numbers. Also , we give relations among the Fibonacci, Lucas and Leonardo numbers. Finally, using the matrix representation of Leonardo numbers, we obtain the some identities of Le...

In this article, we study the Trivariate Fibonacci and Lucas poly-nomials. The classical Tribonacci numbers and Tribonacci polynomials are the special cases of the trivariate Fibonacci polynomials. Also, we obtain some properties of the trivariate Fibonacci and Lucas polynomials. Using these properties, we give some results for the Tribonacci numbe...

In this article, we study the generalized bivariate Fibonacci (GBF) and generalized bivariate Lucas (GBL) polynomials from specifying p(x,y) and?? q(??x,??y), classical bivariate Fibonacci and Lucas polynomials ((p(x,y)=x and q(x,y)=??y). Afterwards, we obtain the some properties of the GBF and GBL polynomials.

In this paper, we prove the q -analogue of the fundamental theorem of Riordan arrays. In particular, by defining two new binary operations *q and *1/q, we obtain a q -analogue of the Riordan representation of the q -Pascal matrix. In addition, by aid of the q -Lagrange expansion formula we get q -Riordan representation for its inverse matrix.

We study an analogue of Riordan representation of Pascal matrices via Fibonomial coefficients. In particular, we establish a relationship between the Riordan array and Fibonomial coefficients, and we show that such Pascal matrices can be represented by an -Riordan pair.

In this paper, we de…ne a generalization of the Pascal matrix with Fontené-Ward generalized binomial coe¢ cients. Also, we obtain inverse of a new matrix n(u). We show that the matrix n(u) can be factorized by some special matrices. Using the factorizations of this matrix, we obtain new combinatorial identities for the Fontené-Ward generalized bino...

In this paper, we study a numerical method based on polynomial approximation, using the shifted Chebyshev polynomial, to construct the approximate solutions of the one dimensional linear Klein-Gordon equation with constant coefficients. Also, we give general forms of the operational matrices of integral and derivative. We solve two illustrative exa...

In this article, we study the bivariate Fibonacci and Lucas p–polynomials (p⩾0 is integer) from which, specifying x, y and p, bivariate Fibonacci and Lucas polynomials, bivariate Pell and Pell-Lucas polynomials, Jacobsthal and Jacobsthal–Lucas polynomials, Fibonacci and Lucas p–polynomials, Fibonacci and Lucas p–numbers, Pell and Pell–Lucas p–numbe...

In this article, we define the m-extension of the Fibonacci and Lucas p-numbers (p⩾0 is integer and m>0 is real number) from which, specifying p and m, classic Fibonacci and Lucas numbers (p=1, m=1), Pell and Pell–Lucas numbers (p=1, m=2), Fibonacci and Lucas p-numbers (m=1), Fibonacci m-numbers (p=1), Pell and Pell–Lucas p-numbers (m=2) are obtain...

In this study, we introduce a new generalization of the second order polynomial sequences. Namely, we define the Horadam polynomials se-quence. Afterwards, we investigate the some properties of the Horadam polynomials.

In this paper we introduce an extension of the hyperbolic Fibonacci and Lucas functions which were studied by A. Stakhov and B. Rozin [Chaos Solitons Fractals 23, No. 2, 379–389 (2005; Zbl 1130.33300)]. Namely, we define hyperbolic functions by the second order recurrence sequences and study their hyperbolic and recurrence properties. We give the c...

In this paper we define the Pell and Pell-Lucas p-numbers and derive analytical formulas for these numbers. These formulas are similar to Binet’s formula for the classical Pell numbers.

In this paper, we define the hyperbolic modified Pell functions by the modified Pell sequence and classical hyperbolic functions. Afterwards, we investigate the properties of the modified Pell functions.

In this paper, we obtain the spectral norm and eigenvalues of circulant matrices with Horadam's numbers. Furthermore, we define the semicirculant matrix with these numbers and give the Euclidean norm of this matrix.

In this paper, we give some properties of the modifled Pell, Jacobsthal and Jacobsthal-Lucas numbers. We then deflne the circulant, nega- cyclic and semicirculant matrices with these numbers and investigate the norms, eigenvalues and determinants of these matrices.