On Hybrid Hyper <i>k</i>-Pell, <i>k</i>-Pell–Lucas, and Modified <i>k</i>-Pell Numbers

Abstract

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell–Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers.

Keywords

• hybrid hyper <i>k</i>-Pell
• hybrid hyper <i>k</i>-Pell–Lucas
• hybrid hyper Modified <i>k</i>-Pell numbers
• generating functions
• the Binet formula
• identities