Emrah Kilic

On the order-k generalized Lucas numbers

In this paper we give a new generalization of the Lucas numbers in matrix representation. Also we present a relation between the generalized order-k Lucas sequences and Fibonacci sequences.

The Binet formula, sums and representations of generalized Fibonacci p-numbers

In this paper, we consider the generalized Fibonacci p-numbers and then we give the generalized Binet formula, sums, combinatorial representations and generating function of the generalized Fibonacci p-numbers. Also, using matrix methods, we derive an explicit formula for the sums of the generalized Fibonacci p-numbers.

A computational algorithm for special nth-order pentadiagonal Toeplitz determinants

Abstract In this short note, we present a fast and reliable algorithm for evaluating special n th-order pentadiagonal Toeplitz determinants in linear time. The algorithm is suited for implementation using Computer Algebra Systems (CAS) such as MACSYMA and MAPLE. Two illustrative examples are given.

On a constant-diagonals matrix

In this paper, we consider a constant-diagonals matrix. The matrix was discussed in Witula and Slota [R. Witula, D. Slota, On computing the determinants and inverses of some special type of tridiagonal and constant-diagonals matrices, Appl. Math. Comput. 189 (1) (2007) 514–527]. The authors gave some results on determinant and the inverse of the matrix for some special cases. We give LU factorization and then compute determinant of the matrix. We determine eigenvalues of the matrix. Further we obtain some relationships between permanent of the matrix and terms of a certain recurrence relation.

The generalized Fibonomial matrix

The Fibonomial coefficients are known as interesting generalizations of binomial coefficients. In this paper, we derive a (k+1)th recurrence relation and generating matrix for the Fibonomial coefficients, which we call generalized Fibonomial matrix. We find a nice relationship between the eigenvalues of the Fibonomial matrix and the generalized right-adjusted Pascal matrix; that they have the same eigenvalues. We obtain generating functions, combinatorial representations, many new interesting identities and properties of the Fibonomial coefficients. Some applications are also given as examples.

More on the infinite sum of reciprocal Fibonacci, Pell and higher order recurrences

Recently [Z. Wenpeng, W. Tingting, Applied Mathematics and Computation 218 (10) (2012) 6164-6167; T. Komatsu, V. Laohakosol, Journal of Integer Sequences 13 (5) (2010) Article 10.5.8.] computed partial infinite sums including reciprocal usual Fibonacci, Pell and generalized order-k Fibonacci numbers. In this paper we will present generalizations of earlier results by considering more generalized higher order recursive sequences with additional one coefficient parameter.

The q-Pilbert matrix

A generalized Filbert matrix is introduced, sharing properties of the Hilbert matrix and Fibonacci numbers. Explicit formulae are derived for the LU-decomposition and their inverses, as well as the Cholesky decomposition. The approach is to use q-analysis and to leave the justification of the necessary identities to the q-version of Zeilbergers celebrated algorithm.

Factorizations and Representations of Binary Polynomial Recurrences by Matrix Methods

Abstract : In this paper, the authors derive factorizations and representations of a polynomial analogue of an arbitrary binary sequence by matrix methods. It generalizes various results on Fibonacci, Lucas, Chebyshev, and Morgan-Voyce polynomials.

On binomial sums for the general second order linear recurrence.

Abstract In this short paper we establish identities involving sums of products of binomial coefficients and coefficients that satisfy the general second–order linear recurrence. We obtain generalizations of identities of Carlitz, Prodinger and Haukkanen.

The generalized q-Pilbert matrix

A generalized q-Pilbert matrix from[KILIÇ, E.-PRODINGER, H.: The q-Pilbert matrix, Int. J. Comput. Math. 89 (2012), 1370–1377] is further generalized, introducing one additional parameter. Explicit formulæ are derived for the LU-decomposition and their inverses, as well as the Cholesky decomposition. The approach is to use q-analysis and to leave the justification of the necessary identities to the q-version of Zeilberger’s celebrated algorithm. However, the necessary identities have appeared already in disguised form in the paper referred above, so that no new computations are necessary.