Properties of hyperbolic generalized Pell numbers

Abstract

In this paper, we introduce the generalized hyperbolic Pell numbers over the bidimensional Clifford algebra of hyperbolic numbers. As special cases, we deal with hyperbolic Pell and hyperbolic Pell–Lucas numbers. We present Binet’s formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan’s, Cassini’s, d’Ocagne’s, Gelin–Cesàro’s, Melham’s identities and present matrices related to these sequences.

Gaussian-bihyperbolic Numbers Containing Pell and Pell-Lucas Numbers

“…They also gave some important features of these newly defined numbers. Soykan and Göcen (Soykan and Göcen, 2020) introduced the generalized hyperbolic Pell numbers over the bidimensional Clifford algebra of hyperbolic numbers. Azak and Güngör (Azak and Güngör, 2017) defined the dual complex Fibonacci and Lucas numbers and gave the wellknown properties for these numbers.…”

Split complex bi-periodic Fibonacci and Lucas numbers

“…The literature contains many articles that related to the special number sequences such as Fibonacci, Lucas, Pell ( [2,3,6,8,14,15,17,18]). One of these articles goes through to the bi-periodic Fibonacci (or, equivalently, generalized Fibonacci) and the bi-periodic Lucas (or, equivalently, generalized Lucas).…”

Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑nk=0 xkWmk+j

“…Pell sequence has been studied by many authors and more detail can be found in the extensive literature dedicated to these sequences, see for example, [22], [23], [24], [25], [26], [27], [28], [29]. For higher order Pell sequences, see [30], [31], [32], [33], [34], [35].…”