Gaussian-Hybrid Numbers Obtained From Pell and Pell-Lucas Sequences

Abstract

In this study, we define a new type of Pell and Pell-Lucas numbers which are called Gaussian-hybrid Pell and Pell-Lucas numbers. We also give negaGaussian-hybrid Pell and Pell-Lucas numbers, the characteristic number and the type number of Gaussian-hybrid Pell and Pell-Lucas numbers. Also, some sum ve product properties of Pell and Pell-Lucas numbers are given. Moreover, we obtain the Binet’s formula, generating function formula, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity and some sum formulas for the Gaussian-hybrid Pell and Pell-Lucas numbers. Some algebraic proporties of Gaussian-hybrid Pell and Pell-Lucas numbers are investigated. Futhermore, we give the matrix representation of Gaussian-hybrid Pell and Pell-Lucas numbers.

On Generalized Tetranacci Numbers: Closed Forms of the Sum Formulas \(\sum_{k=0}^{n}kx^{k}W_{k}\) and \(\sum_{k=1}^{n}kx^{k}W_{-k}\)

“…Papers which deal with sum formulas Pell and Pell-Lucas [12][13][14][15][16] Generalized Fibonacci [17][18][19][20][21][22][23] Generalized Tribonacci [24][25][26][27] Generalized Tetranacci [6,24,28,29] Generalized Pentanacci [24,30,31] Generalized Hexanacci [32,33] The following theorem present some linea sum formulas of generalized Tetranacci numbers with positive subscripts.…”

On New Polynomial Sequences Constructed to Each Vertex in an n‐Gon

“…In this direction, many researchers work on it and study its wings, and the Lucas numbers became a vital role for them to study as can be seen in [1][2][3][4][5][6][7][8]. While framing these sequences, other sequences came into existence such as Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas by replacing the initial conditions [9][10][11][12][13][14][15].…”