Abstract:We give a new definition of Narayana polynomials and show that there is a relationship between the coefficient of the new Narayana polynomials and Pascal’s triangle. We define the Gauss Narayana numbers and their polynomials. Then we show that there is a relationship between the Gauss Narayana polynomials and the new Narayana polynomials. Also, we show that there is a relationship between the derivatives of the new Narayana polynomials and Pascal’s triangle. We also explain the relationship between the new Nar… Show more
“…Horadam defined the complex Fibonacci numbers in 1963. Since then, many authors investigated the Gaussian shapes of these sequences and their properties; see for example [6][7][8][9][10][11][12][13][14][15][16]. In [8], the Gaussian forms of the Fibonacci and Lucas number sequences are given recurrently by…”
Bu çalışmanın amacı Gauss Bronz Lucas sayı dizisini tanıtmak ve incelemektir. İlk olarak Bronz Lucas sayılarını genişleterek Gauss Bronz Lucas sayılarını tanımladık. Daha sonra bu sayı dizisi için Binet formülü ve üreteç fonksiyonunu bulduk. Ayrıca Gauss Bronz Lucas sayıları ile ilgili bazı toplam formülleri ve matrisleri araştırdık. Son olarak, bu dizinin Binet formülünü dikkate alarak Catalan, Cassini ve d’Ocagne özdeşlikleri gibi bilinen eşitlikleri elde ettik.
“…Horadam defined the complex Fibonacci numbers in 1963. Since then, many authors investigated the Gaussian shapes of these sequences and their properties; see for example [6][7][8][9][10][11][12][13][14][15][16]. In [8], the Gaussian forms of the Fibonacci and Lucas number sequences are given recurrently by…”
Bu çalışmanın amacı Gauss Bronz Lucas sayı dizisini tanıtmak ve incelemektir. İlk olarak Bronz Lucas sayılarını genişleterek Gauss Bronz Lucas sayılarını tanımladık. Daha sonra bu sayı dizisi için Binet formülü ve üreteç fonksiyonunu bulduk. Ayrıca Gauss Bronz Lucas sayıları ile ilgili bazı toplam formülleri ve matrisleri araştırdık. Son olarak, bu dizinin Binet formülünü dikkate alarak Catalan, Cassini ve d’Ocagne özdeşlikleri gibi bilinen eşitlikleri elde ettik.
“…(2) where (3) The Narayana numbers and their properties have been studied by Özkan, Ramirez, Petersen, et al [3][4][5][6][7][8][9][10]. In particular, Petersen [5] places them in the Euler-Macmahon-Carlitz/Riordan combinatorial spectrum.…”
In this paper, inspiring Hosoya’s triangle, we define a new Narayana triangle. Then, we represent this Narayana triangle geometrically on the plane. In addition, we give some identities and properties of the new Narayana triangle.
“…One of the most well-known number sequences is the Jacobsthal numbers [4,15,19]. One of the most important of these generalizations is those about Gaussian [13,14,16,18]. Özkan et al defined Gauss Fibonacci polynomials, Gauss Lucas polynomials and gave their applications in [12].…”
In this paper, we define d− Gaussian Jacobsthal polynomials and d−Gaussian Jacobsthal-Lucas polynomials. We present the sum, generating functions and Binet formulas of these polynomials. We give the matrix representations of them. We present these matrices as binary representation according to the Riordan group matrix representation. By using Riordan method, we give factorizations of Pascal matrix involving d−Gaussian Jacobsthal polynomials and d−Gaussian Jacobsthal-Lucas polynomials. We give the inverse of matrices of these polynomials.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.