Emrah Kılıç

Bio: Emrah Kılıç is an academic researcher from TOBB University of Economics and Technology. The author has contributed to research in topics: Fibonacci number & Matrix (mathematics). The author has an hindex of 20, co-authored 129 publications receiving 1125 citations. Previous affiliations of Emrah Kılıç include Mansoura University & Stellenbosch University.

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

Abstract: 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.

Tribonacci sequences with certain indices and their sums

Abstract: In this paper, we derive new recurrence relations and generating matrices for the sums of usual Tribonacci numbers and 4n subscripted Tribonacci sequences, fT4ng ; and their sums. We obtain explicit formulas and combinatorial representations for the sums of terms of these sequences. Finally we represent relationships between these sequences and permanents of certain matrices. 1. Introduction The Tribonacci sequence is de…ned by for n > 1 Tn+1 = Tn + Tn 1 + Tn 2 where T0 = 0; T1 = 1; T2 = 1: The few …rst terms are 0; 1; 1; 2; 4; 7; 13; 24; 44; 81; 149; : : : : We de…ne Tn = 0 for all n 0: The Tribonacci sequence is a well known generalization of the Fibonacci sequence. In (see page 527-536, [3]), one can …nd some known properties of Tribonacci numbers. For example, the generating matrix of fTng is given by Q = 24 1 1 1 1 0 0 0 1 0 35n = 24 Tn+1 Tn + Tn 1 Tn Tn Tn 1 + Tn 2 Tn 1 Tn 1 Tn 2 + Tn 3 Tn 2 35 : For further properties of Tribonacci numbers, we refer to [1, 4, 5]. Let Sn = Pn k=0 Tk: (1.1) In this paper, we obtain generating matrices for the sequences fTng,fT4ng, fSng and fS4ng : (The second result follows from a third order recurrence for T4n:) We also obtain Binet-type explicit and closed-form formulas for Sn and S4n: Further on, we present relationships between permanents of certain matrices and all the above-mentioned sequences. 2000 Mathematics Subject Classi…cation. 11B37, 15A36, 11P.

On the Generalized Order-$k$ Fibonacci and Lucas Numbers

Abstract: In this paper we consider the generalized order-k Fibonacci and Lucas numbers. We give the generalized Binet formula, combinatorial representation and some relations involving the generalized order-k Fibonacci and Lucas numbers.

Hybrid threshold adaptable quantum secret sharing scheme with reverse Huffman-Fibonacci-tree coding

Abstract: With prevalent attacks in communication, sharing a secret between communicating parties is an ongoing challenge. Moreover, it is important to integrate quantum solutions with classical secret sharing schemes with low computational cost for the real world use. This paper proposes a novel hybrid threshold adaptable quantum secret sharing scheme, using an m-bonacci orbital angular momentum (OAM) pump, Lagrange interpolation polynomials, and reverse Huffman-Fibonacci-tree coding. To be exact, we employ entangled states prepared by m-bonacci sequences to detect eavesdropping. Meanwhile, we encode m-bonacci sequences in Lagrange interpolation polynomials to generate the shares of a secret with reverse Huffman-Fibonacci-tree coding. The advantages of the proposed scheme is that it can detect eavesdropping without joint quantum operations, and permits secret sharing for an arbitrary but no less than threshold-value number of classical participants with much lower bandwidth. Also, in comparison with existing quantum secret sharing schemes, it still works when there are dynamic changes, such as the unavailability of some quantum channel, the arrival of new participants and the departure of participants. Finally, we provide security analysis of the new hybrid quantum secret sharing scheme and discuss its useful features for modern applications.

Applied Mathematics and Computation

Abstract: The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Hybrid threshold adaptable quantum secret sharing scheme with reverse Huffman-Fibonacci-tree coding

Abstract: With prevalent attacks in communication, sharing a secret between communicating parties is an ongoing challenge. Moreover, it is important to integrate quantum solutions with classical secret sharing schemes with low computational cost for the real world use. This paper proposes a novel hybrid threshold adaptable quantum secret sharing scheme, using an m-bonacci orbital angular momentum (OAM) pump, Lagrange interpolation polynomials, and reverse Huffman-Fibonacci-tree coding. To be exact, we employ entangled states prepared by m -bonacci sequences to detect eavesdropping. Meanwhile, we encode m -bonacci sequences in Lagrange interpolation polynomials to generate the shares of a secret with reverse Huffman-Fibonacci-tree coding. The advantages of the proposed scheme is that it can detect eavesdropping without joint quantum operations, and permits secret sharing for an arbitrary but no less than threshold-value number of classical participants with much lower bandwidth. Also, in comparison with existing quantum secret sharing schemes, it still works when there are dynamic changes, such as the unavailability of some quantum channel, the arrival of new participants and the departure of participants. Finally, we provide security analysis of the new hybrid quantum secret sharing scheme and discuss its useful features for modern applications.