Yazar "Irmak, Nurettin" seçeneğine göre listele

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    Balancing Diophantine triples with distance 1
    (SPRINGER, 2015) Alp, Murat; Irmak, Nurettin; Szalay, Laszlo
    For a positive real number let the Balancing distance be the distance from to the closest Balancing number. The Balancing sequence is defined by the initial values , and by the binary recurrence relation , . In this paper, we show that there exist only one positive integer triple such that the Balancing distances , and all are exactly 1.
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    BALANCING WITH BALANCING POWERS
    (UNIV MISKOLC INST MATH, 2013) Irmak, Nurettin
    In this paper, the Diophantine equation B-1(k) + B-2(k) + ... + B-n-1(k) = B-n+1(k) + B-n+2(k) + ... + B-n+r(l) + for the positive integer unknowns n >= 2, k, l and r is studied in certain cases, where B-n denotes the nth term of the balancing
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    BALANCING WITH POWERS OF THE LUCAS SEQUENCE OF RECURRENCE un = Aun-1 - un-2
    (Univ Miskolc Inst Math, 2017) Irmak, Nurettin; Alp, Murat
    In this paper, we show that there is no solution of the diophantine equation u(1)(k) + u(2)(k) + . . . + u(n+1)(l) + u(n+2)(l) + . . . + u(n+r)(l) for special cases of k and l where the elements of sequence {u(n)} satisfy the relation u(n) = Aun-1 - u(n-2) with u(0) = 0, u(1) = 1 and A >= 3 is a positive integer.
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    BINOMIAL IDENTITIES INVOLVING THE GENERALIZED FIBONACCI TYPE POLYNOMIALS
    (CHARLES BABBAGE RES CTR, 2011) Kilic, Emrah; Irmak, Nurettin
    We present some binomial identities for sums of the bivariate Fibonacci polynomials and for weighted sums of the usual Fibonacci polynomials with indices in arithmetic progression.
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    Binomial identities involving the generalized fibonacci type polynomials
    (Charles Babbage Research Centre, 2011) Kilic, Emrah; Irmak, Nurettin
    We present some binomial identities for sums of the bivariate Fibonacci polynomials and for weighted sums of the usual Fibonacci polynomials with indices in arithmetic progression.
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    Decompositions of the Cauchy and Ferrers-Jackson polynomials
    (UNIV OSIJEK, DEPT MATHEMATICS, 2016) Irmak, Nurettin; Kilic, Emrah
    Recently, Witula and Slota have given decompositions of the Cauchy and Ferrers-Jackson polynomials [Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences, Central Europan J. Math., 2006]. Our main purpose is to derive a different decomposition of the Cauchy and Ferrers-Jackson polynomials. Our approach is to use the Waring formula and the Saalschutz identity to prove the claimed results. Also, we obtain generalizations of the results of Carlitz, Hunter and Koshy as corollaries of our results about sums and differences of powers of the Fibonacci and Lucas numbers.
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    DIOPHANTINE TRIPLES AND REDUCED QUADRUPLES WITH THE LUCAS SEQUENCE OF RECURRENCE
    (CROATIAN MATHEMATICAL SOC, 2014) Irmak, Nurettin; Szalay, Laszlo
    In this study, we show that there is no positive integer triple {a, b, c} such that all of ab+1, ac+1 and bc+1 are in the sequence {u(n)}n= 0 satisfies the recurrence un=Aun-1-un-2 with the initial values u0=0, u1=1. Further, we investigate the analogous question for the quadruples {a,b,c,d} with abc+1=ux, bcd+1=uy, cda+1=uz and dab+1=ut, and deduce the non-existence of such quadruples.
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    Factorial-like values in the balancing sequence
    (Univ Osijek, Dept Mathematics, 2018) Irmak, Nurettin; Liptai, Kalman; Szalay, Laszlo
    In this paper, we solve a few Diophantine equations linked to balancing numbers and factorials. The basic problem consists of solving the equation B-y = x! in positive integers x, y, which has only one nontrivial solution B-2 = 6 = 3!, as a direct consequence of the theorem of F. Luca [5]. A more difficult problem is to solve B-y = x(2)!/x(1)!, but we were able to handle it under some conditions. Two related problems are also studied.
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    FAREY-PELL SEQUENCE, APPROXIMATION TO IRRATIONALS AND HURWITZ'S INEQUALITY
    (INT CENTER SCIENTIFIC RESEARCH & STUDIES, 2016) Akkus, Ilker; Irmak, Nurettin; Kizilaslan, Gonca
    The 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 given
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    Fibonacci ve Lucas sayılarının ters toplamları ve uygulamarı
    (Niğde Üniversitesi, 2010) Irmak, Nurettin; Kılıç, Emrah; Atay, Mehmet Tarık
    Bu çalışmada Fibonacci ve Lucas sayılarının bazı özel alt dizilerini göz önüne alınarak ters toplamlarına ait formüller elde edilmiştir. Ters toplamlarına ait formüller elde etmek için, üreteç fonksiyonları ve alt dizilerinin indirgeme bağıntısı kullanılmıştır. Bu tez ile ters toplamlara ait önceki formülleri genelleştirilmiştir.
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    Fuzzy Fibonacci and Fuzzy Lucas Numbers with their Properties
    (2019) Irmak, Nurettin; Demirtaş, Naime
    In this paper, we combine the important concepts which are Fuzzy numbers and Fibonacci, Lucasnumbers. We introduce the concepts of Fuzzy Fibonacci and Fuzzy Lucas numbers by this combination.By this motivation, we provide a bridge between the areas Fuzzy sets and number theory. Afterwards,we generalize their well-known properties by the definitions of Fuzzy Fibonacci and Lucas numbers.
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    INCOMPLETE BALANCING AND LUCAS-BALANCING NUMBERS
    (EDITURA ACAD ROMANE, 2018) Patel, Bijan Kumar; Irmak, Nurettin; Ray, Prasanta Kumar
    The aim of this article is to establish some combinatorial expressions of balancing and Lucas-balancing numbers and investigate some of their properties.
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    Incomplete balancing and lucas-balancing numbers
    (Editura Academiei Romane, 2018) Kumar Patel, Bijan; Irmak, Nurettin; Kumar Ray, Prasanta; Zaharescu, Alexandru
    The aim of this article is to establish some combinatorial expressions of balancing and Lucas-balancing numbers and investigate some of their properties. © 2018 Editura Academiei Romane. All Rights Reserved.
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    Lineer reküransların diyafont üçlüsü
    (Niğde Üniversitesi / Fen Bilimleri Enstitüsü, 2014) Irmak, Nurettin; Alp, Murat
    Bu çalışma altı bölümden oluşmaktadır. Birinci bölümde, lineer reküransların tanımı, ikinci mertebeden özel lineer reküranslar, diyafont m-lisinin tanımı ve konu ile ilgili literatür verilmiştir. İkinci bölümde, balans diyafont üçlüsü tanımlanıp, balans diyafont üçlüsünün olmadığı gösterilmiştir. Üçüncü bölümde, u_0=0 ve u_1=1 başlangıç koşulları altında, u_n=A*u_(n-1)-u_(n-2) lineer reküransını sağlayan dizinin diyafont üçlüsünün olmadığı gösterilmiştir. Dördüncü bölümde, k-periyodik dizilerin tanımı verilmiş ve dizi elemanları ile ilgili Binet formülleri elde edilmiştir. Beşinci bölümde, dördüncü mertebeden pellans sayı dizisi tanımlanmış ve bu dizinin diyafont üçlüsünün olmadığı incelenmiştir. Son bölümde, yapılan çalışmaların literatüre katkıları ve bu konudaki önerileri verilmiştir.
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    Lucas numbers of the form (k2t)
    (Univ Tartu Press, 2019) Irmak, Nurettin; Szalay, Laszlo
    Let L-m denote the m(th) Lucas number. We show that the solutions to the diophantine equation ((2t)(k)) = L-m, in non-negative integers t, k <= 2(t-1), and m, are (t, k, m) = (1, 1, 0), (2,1, 3), and (a, 0,1) with non-negative integers a.
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    More Identities for Fibonacci and Lucas Octonions
    (Murat TOSUN, 2020) Irmak, Nurettin; Açıkel, Abdullah
    In this paper, we give a new approach to obtain identities for Fibonacci and Lucas octonions. © MSAEN.
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    MORE IDENTITIES FOR FIBONACCI AND LUCAS QUATERNIONS
    (Ankara Univ, Fac Sci, 2020) Irmak, Nurettin
    In this paper, we define the associate matrix as F = [GRAPHICS] . By the means of the matrix F, we give several identities about Fibonacci and Lucas quaternions by matrix methods. Since there are two different determinant definitions of a quaternion square matrix (whose entries are quaternions), we obtain different Cassini identities for Fibonacci and Lucas quaternions apart from Cassini identities that given in the papers [5] and [7].
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    On factorials in Perrin and Padovan sequences
    (Tubitak Scientific & Technological Research Council Turkey, 2019) Irmak, Nurettin
    Assume that w(n) is the n th term of either Padovan or Perrin sequence. In this paper, we solve the equation w(n) = m! completely.
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    On k-periodic binary recurrences
    (E K F Liceum Kiado, 2012) Irmak, Nurettin; Szalay, Laszlo
    We apply a new approach, namely the fundamental theorem of homogeneous linear recursive sequences, to k-periodic binary recurrences which allows us to determine Binet's formula of the sequence if k is given. The method is illustrated in the cases k = 2 and k = 3 for arbitrary parameters. Thus we generalize and complete the results of Edson-Yayenie, and Yayenie linked to k = 2 hence they gave restrictions either on the coefficients or on the initial values. At the end of the paper we solve completely the constant sequence problem of 2-periodic sequences posed by Yayenie.
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    On repdigits as product of consecutive Lucas numbers
    (Bulgarian Acad Science, 2018) Irmak, Nurettin; Togbe, Alain
    Let (L-n)(n >= 0 )be the Lucas sequence. D. Marques and A. Togbe [7] showed that if F-n . . . Fn+k-1 is a repdigit with at least two digits, then (k, n) = (1, 10), where (F-n)(>= 0) is the Fibonacci sequence. In this paper, we solve the equation L-n . . . Ln+k-1 = a (10(m) - 1/9) , where 1 <= a <= 9, n, k >= 2 and in are positive integers.