On the new Narayana polynomials, the Gauss Narayana numbers and their polynomials

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 Narayana polynomials and the known Pell numbers. Finally, we give the Hankel transform of the new Narayana polynomials.

Gaussian Bronze Lucas Numbers

“…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…”

New Narayana Triangle

“…(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.…”

D- Gaussian Jacobsthal, D- Gaussian Jacobsthallucas Polynomials and Their Matrix Representations E. Ozkan, M. Uysal

“…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].…”