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Öğe A unified approach to generalized Pascal-like matrices: q-analysis(Elsevier Inc., 2023) Akkus, Ilker; Kizilaslan, Gonca; Verde-Star, LuisIn this paper, we present a general method to construct q-analogues and other generalizations of Pascal-like matrices. Our matrices are obtained as functions of strictly lower triangular matrices and include several types of generalized Pascal-like matrices and matrices related with modified Hermite polynomials of two variables and other polynomial sequences. We find explicit expressions for products, powers, and inverses of the matrices and also some factorization formulas using this method. © 2023 Elsevier Inc.Öğe A curious matrix-sum identity and certain finite sums identities(World Scientific Publ Co Pte Ltd, 2015) Kilic, Emrah; Akkus, Ilker; Omur, Nese; Ulutas, Yucel T.In this paper, we consider two generalized binary sequences and then give a generalization of a matrix equality proposed as an advanced problem. Then, we derive new certain finite sums including the generalized binary sequences as applications.Öğe Diophantine Equations Related with Linear Binary Recurrences(Tarbiat Modares Univ, Acecr, 2022) Kilic, Emrah; Akkus, Ilker; Omur, NeseIn this paper we find all solutions of four kinds of the Dio-phantine equations x(2) +/- V(t)xy - y(2) +/- x = 0 and x(2) +/- V(t)xy - y(2) +/- y = 0, for an odd number t, and, x(2) +/- V(t)xy + y(2) - x = 0 and x(2) +/- V(t)xy +/- y(2) - y = 0, for an even number t, where V-n is a generalized Lucas number. This paper continues and extends a previous work of Bahramian and Daghigh.Öğe Farey-Pell Sequence, Approximation To Irrationals And Hurwitz's Inequality(Int Center Scientific Research & Studies, 2016) Akkus, Ilker; Irmak, Nurettin; Kizilaslan, GoncaThe purpose of this paper is to give the notion of Farey-Pell sequence. We investigate some identities of the Farey-Pell sequence. Finally, a generalization of Farey-Pell sequence and an approximation to irrationals via Farey-Pell fractions are givenÖğe The Fibonacci Octonions(Springer Basel Ag, 2015) Kecilioglu, Osman; Akkus, IlkerIn the present paper, we introduce the Fibonacci and Lucas octonions and give the generating function and Binet formulae for these octionions. In addition, we give some identities and properties of them.Öğe Formulas for binomial sums including powers of Fibonacci and Lucas numbers(Univ Politehnica Bucharest, Sci Bull, 2015) Kilic, Emrah; Akkus, Ilker; Omur, Nese; Ulutas, Yucel TurkerRecently Prodinger [2] proved general expansion formulas for sums of powers of Fibonacci and Lucas numbers. In this paper, we will prove general expansion formulas for binomial sums of powers of Fibonacci and Lucas numbers.Öğe Generalization of a statistical matrix and its factorization(Taylor & Francis Inc, 2019) Akkus, Ilker; Kizilaslan, GoncaWe consider a special matrix with integer coefficients and obtain an LU factorization for its member by giving explicit closed-form formulae of the entries of L and U. Our result is applied to give the closed-form formula of the inverse of the considered matrix. We give the relation between the defined matrix and Helmert matrix which has been used for proving the statistical independence of a number of statistics. Also we find the condition numbers of some matrices for some special values of q.Öğe Generalized Binomial Convolution of the mth Powers of the Consecutive Integers with the General Fibonacci Sequence(De Gruyter Open Ltd, 2016) Kilic, Emrah; Akkus, Ilker; Omur, Nese; Ulutas, Yucel T.In this paper, we consider Gauthier's generalized convolution and then define its binomial analogue as well as alternating binomial analogue. We formulate these convolutions and give some applications of them.Öğe The Lehmer matrix with recursive factorial entries(Academic Publication Council, 2015) Akkus, IlkerA generalized Lehmer matrix with recursive entries from Kilic et al. (2010b) is further generalized, introducing three additional parameters and taking recursive factorials instead of a term. Certain formulae are derived for the LU and Cholesky factorizations and their inverses, as well as the determinants. Then we precisely compute the elements of the inverse of the generalized Lehmer matrix.Öğe The linear algebra of a Pascal-like matrix(TAYLOR & FRANCIS LTD, 2020) Zheng, De-Yin; Akkus, Ilker; Kizilaslan, GoncaA Pascal-like matrix is constructed whose row entries are the terms of the modified Hermite polynomials of two variables. The multiplication of the two Pascal-like matrices and the power and inverse of the Pascal-like matrix have very similar results to the well-known Binomial Matrix, and the factorization of the Pascal-like matrix has also been given, which is completely similar to the factorization of Binomial Matrix. Furthermore, two simple examples are given to show the application of the power and factorization properties of the Pascal-like matrix. Finally, a generalization of the Pascal-like matrix is given and some combinatorial identities are obtained such as Tepper-like identity.Öğe New Curiosity Bivariate Quadratic Quaternionic Polynomials and Their Roots(Ovidius Univ Press, 2021) Akkus, Ilker; Kizilaslan, GoncaWe consider the second-order linear homogeneous quaternion recurrence solutions for some new curiosity bivariate quadratic quaternionic equations.Öğe New Filbert and Lilbert matrices with asymmetric entries(WALTER DE GRUYTER GMBH, 2020) Bozdag, Hacer; Kilic, Emrah; Akkus, IlkerIn this paper, two new analogues of the Hilbert matrix with four-parameters have been introduced. Explicit formul ae are derived for the LU-decompositions and their inverses, and the inverse matrices of these analogue matrices. (C) 2020 Mathematical Institute Slovak Academy of SciencesÖğe NEW REPRESENTATIONS OF PASCAL MATRIX VIA OPERATIONAL q-CALCULUS(Honam Mathematical Soc, 2022) Zheng, De-Yin; Akkus, Ilker; Kizilaslan, GoncaIn this paper we introduce two type of representations of the Pascal matrix via induced transformations of some q-derivatives as well as their some combinatorial applications.Öğe On Fibonomial Sums Identities With Special Sign Functions: Analytically Q-Calculus Approach(Walter De Gruyter Gmbh, 2018) Kilic, Emrah; Akkus, IlkerRecently Marques and Trojovsky [On some new identities for the Fibonomial coefficients, Math. Slovaca 64 (2014), 809-818] presented interesting two sum identities including the Fibonomial coefficients and Fibonacci numbers. These sums are unusual as they include a rare sign function and their upper bounds are odd. In this paper, we give generalizations of these sums including the Gaussian q-binomial coefficients. We also derive analogue q-binomial sums whose upper bounds are even. Finally we give q-binomial sums formula whose weighted functions are different from the earlier ones. To prove the claimed results, we analytically use q-calculus. (C) 2018 Mathematical Institute Slovak Academy of SciencesÖğe On some Properties of Tribonacci Quaternions(Ovidius Univ Press, 2018) Akkus, Ilker; Kizilaslan, GoncaIn this paper, we give some properties of the Tribonacci and Tribonacci-Lucas quaternions and obtain some identities for them.Öğe Partial sums of the Gaussian q-binomial coefficients, their reciprocals, square and squared reciprocals with applications(Univ Miskolc Inst Math, 2019) Kilic, Emrah; Akkus, IlkerIn this paper, we shall derive formulae for partial sums of the Gaussian q-binomial coefficients, their reciprocals, squares and squared reciprocals. To prove the claimed results, we use q-calculus. As applications of our results, we give some interesting generalized Fibonomial sums formulae.Öğe Quaternions: Quantum calculus approach with applications(Academic Publication Council, 2019) Akkus, Ilker; Kizilaslan, GoncaIn this paper we introduce two types of quaternion sequences with components including quantum integers. We also introduce quantum quaternion polynomials. Moreover, we give some properties and identities for these quantum quaternions and polynomials. Finally, we give time evolution and rotation applications for some specific quaternion sequences. The applications can be converted into quantum integer forms under suitable conditions with similar considerations.Öğe Some Generalized Fibonomial Sums related with the Gaussian q-Binomial sums(Soc Matematice Romania, 2012) Kilic, Emrah; Akkus, Ilker; Ohtsuka, HideyukiIn this paper, we consider some generalized Fibonomial sums formulae and then prove them by using the Cauchy binomial theorem and q-Zeilberger algorithm in Mathematica session.Öğe Some New Quaternionic Quadratics with Zeros in Terms of Second Order Quaternion Recurrences(Springer Basel Ag, 2019) Akkus, Ilker; Kizilaslan, GoncaIn this paper a comprehensive analysis of the Horadam quaternion zeros for some new types of bivariate quadratic quaternion polynomial equations is presented.Öğe Split Fibonacci and Lucas Octonions(Springer Basel Ag, 2015) Akkus, Ilker; Kecilioglu, OsmanIn this paper, split Fibonacci and Lucas octonions are proposed and their some properties and relations are obtained.
