Q-Analogues of biperiodic fibonacci and lucas sedenions

Abstract

Quantum calculus has an important areas in many areas such as number theory, physics and mathematics. In this paper, by taking some useful notations from quantum calculus, we define the biperiodic Fibonacci and Lucas sedenions based on a q-parameter. There are many studies in the literature on the biperiodic Fibonacci and Lucas sequences. Apart from the earlier studies, we define q-analogues of the biperiodic Fibonacci and Lucas sequences. Then, we present qanalogues of the biperiodic Fibonacci and Lucas sedenions. We also investigate the generating functions and the Binet formulas of these sedenions. In addition, we give Catalan identity, Cassini identity, d'Ocagne identity and sum binomial formulas of the biperiodic q-Fibonacci and q-Lucas sedenions. Since this study covers the previous studies on Fibonacci and Lucas sequences, it deals with the subject from a wider perspective.