On the Harmonic and Hyperharmonic Fibonacci Numbers

In this paper, we study the theory of the harmonic and the hyperharmonic Fibonacci numbers. Also, we get some combinatoric identities like as harmonic and hyperharmonic numbers and we obtain some useful formulas for $\mathbb{F}_{n}$, which is finite sums of reciprocals of Fibonacci numbers. We obtain spectral and Euclidean norms of circulant matrices involving harmonic and hyperharmonic Fibonacci numbers

Parametric kinds of generalized Apostol-Bernoulli polynomials and their properties

The purpose of this paper is to define generalized Apostol--Bernoulli polynomials with including a new cosine and sine parametric type of generating function using the quasi-monomiality properties and trigonometric functions. In this study, the Apostol-Bernoulli polynomials with three variable are defined with two new generating functions cosine and sine parameters. Then, we investigate multiplicative and derivative operators, diffrential equations, some summation formulas and partial differential equations for these polynomials. Moreover, we introduce Gould--Hopper--Apostol--Bernoulli type polynomials, Hermite--Appell--Apostol--Bernoulli type polynomials and truncated exponential Apostol--Bernoulli type polynomials. Finally, the special cases of these new polynomials are investigated, and the corresponding results are expressed.Comment: 16 page

The proporties of gauss binomial coefficients and fibonomial coefficients

Bu çalışmada Gauss Binomiyel katsayıların ve Fibonomiyel Katsayıların tanımları verildi. Gauss Binomiyel Katsayıların temel özellikleri incelendi. Binomiyel Katsayıları içeren bazı özdeşliklerin q benzerleri verildi. Fibonomiyel Katsayıların temel özellikleri incelendi. Gauss Binomiyel Katsayıların Fibonomiyel Katsayılar cinsinden yazılabileceği ve bazı toplamsal q özdeşliklerin Fibonomiyel Katsayılar cinsinden yazılabileceği gösterildi.In this thesis, the definitions of Gauss Binomial Coefficients and Fibonomial Coefficients were given. Basic features of Gauss Binomial Coefficients were examined. q synonyms of some identities possessing Binomial Coefficients were given. Basic features of Fibonomial Coefficients were examined. It has been proved that Gauss Binomial Coefficients can be expressed in terms of Fibonomial Coefficients and that certain q identities can be expressed in terms of Fibonomial Coefficients

Quadra Lucas-Jacobsthal Sayıları Üzerine

In this paper, we define the Quadra Lucas-Jacobsthal numbers and then, we give some properties of this sequences. Moreover, we obtain spectral norms of circulant matrices with Quadra Lucas-Jacobsthal numbers. 2010 AMS-Mathematical Subject Classification Number: 11B39, 05A19, 15A60Bu makalede, Quadra Lucas-Jacobsthal sayıları tanımlanmış ve bu dizilerin çeşitli özellikleri verilmiştir. Dahası elamanları Quadra Lucas-Jacobsthal sayıları olan sirkülant matrislerin spektral normları elde edilmiştir

On the Bicomplex Generalized Tribonacci Quaternions

In this paper, we introduce the bicomplex generalized tribonacci quaternions. Furthermore, Binet’s formula, generating functions, and the summation formula for this type of quaternion are given. Lastly, as an application, we present the determinant of a special matrix, and we show that the determinant is equal to the n th term of the bicomplex generalized tribonacci quaternions

New Properties and Identities for Fibonacci Finite Operator Quaternions

In this paper, with the help of the finite operators and Fibonacci numbers, we define a new family of quaternions whose components are the Fibonacci finite operator numbers. We also provide some properties of these types of quaternions. Moreover, we derive many identities related to Fibonacci finite operator quaternions by using the matrix representations

A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant

In this paper, the authors present a closed formula for the Horadam polynomials in terms of a tridiagonal determinant and, as applications of the newly-established closed formula for the Horadam polynomials, derive closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the first kind in terms of tridiagonal determinants

New Properties and Identities for Fibonacci Finite Operator Quaternions

In this paper, with the help of the finite operators and Fibonacci numbers, we define a new family of quaternions whose components are the Fibonacci finite operator numbers. We also provide some properties of these types of quaternions. Moreover, we derive many identities related to Fibonacci finite operator quaternions by using the matrix representations

New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers

In this paper, firstly the definitions of the families of three-variable polynomials with the new generalized polynomials related to the generating functions of the famous polynomials and numbers in literature are given. Then, the explicit representation and partial differential equations for new polynomials are derived. The special cases of our polynomials are given in tables. In the last section, the interesting applications of these polynomials are found