Some aspects of Fibonacci polynomial congruences

This paper formulates a definition of Fibonacci polynomials which is slightly different from the traditional definitions, but which is related to the classical polynomials of Bernoulli, Euler and Hermite. Some related congruence properties are developed and some unanswered questions are outlined. Keywords: Congruences, recurrence relations, Fibonacci sequence, Lucas sequences, umbral calculus

Fibonacci toplamları ve fibonacci polinomları

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Fibonacci Bu çalısmada Fibonacci toplamları ve Fibonacci polinomları ele alındı. Birincibölümde konuyla ilgili temel tanımlar ve teoremler verildi. ?kinci bölümde Fibonaccive Lucas polinomları tanıtıldı ve bunlarla ilgili teoremler ifade edildi. Son bölümdede Fibonacci ve Lucas sayılarını katsayı kabul eden polinomlar ele alındı ve türevyardımıyla Fibonacci ve Lucas sayıları ile ilgili toplamlar elde edildi.FibonacciIn this study, Fibonacci polynomials and Fibonacci summations are examined. In thefirst chapter, the main definitions and theorems are given. In the second chapter,Fibonacci and Lucas polynomials are investigated and some theorems concerningwith Fibonacci and Lucas polynomials are given. The last chapter is related to thesummations containing Fibonacci and Lucas numbers

The Measure of the Orthogonal Polynomials Related to Fibonacci Chains: The Periodic Case

The spectral measure for the two families of orthogonal polynomial systems related to periodic chains with N-particle elementary unit and nearest neighbour harmonic interaction is computed using two different methods. The interest is in the orthogonal polynomials related to Fibonacci chains in the periodic approximation. The relation of the measure to appropriately defined Green's functions is established.Comment: 19 pages, TeX, 3 scanned figures, uuencoded file, original figures on request, some misprints corrected, tbp: J. Phys.

Hybrinomials Related to Hyper-Fibonacci and Hyper-Lucas Numbers

ybrid number system is a generalization of complex, hyperbolic and dual numbers. Hybrid numbers and hybrid polynomials have been the subject of much research in recent years. In this paper, hybrinomials related to hyper-Fibonacci and hyper-Lucas numbers are defined. Then some algebraic properties of newly defined hybrinomials are examined such as the recurrence relations and summation formulas. Also, the relation between hybrinomials related to hyper-Fibonacci and hyper-Lucas numbers is given. Additionally, hybrid hyper-Fibonacci and hybrid hyper-Lucas numbers are defined by using the hybrinomials related to hyper-Fibonacci and hyper-Lucas numbers

Chebyshev Polynomials and Fibonacci Numbers

The Chebyshev polynomials arise in several mathematical contexts such as approximation theory, numerical integration, and differential equations. Here we study a combinatorial interpretation of Chebyshev polynomials due to Shapiro, and we use it to give a slight variation of a combinatorial proof of Binet\u27s Formula due to Benjamin, Derks and Quinn. Another beautiful formula for the Fibonacci numbers involves complex roots of unity. Presently, no combinatorial proof is known. We give combinatorial proofs of some related identities as progress toward a full combinatorial proof