Scholarlink

<p>Quantum calculus, which can be considered a generalization of classical calculus, is essential for many areas of mathematics. Recently, the work on quantum calculus and its applications to mathematics has increased significantly. This article investigates the connection between higher-dimensional algebraic structures and quantum calculus by considering <i>q</i>-biperiodic Fibonacci and <i>q</i>-biperiodic Lucas sequences in Cayley–Dickson algebras. In this article, we examine new relations between Cayley–Dickson algebras, which generalize complex numbers to higher dimensions by systematic doubling, and <i>q</i>-biperiodic Fibonacci and <i>q</i>-biperiodic Lucas sequences, using some useful notations from quantum calculus. We also present algebraic properties of <i>q</i>-biperiodic Fibonacci and <i>q</i>-biperiodic Lucas <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(2^k\)</EquationSource> </InlineEquation>-ons, binomial sums, generating functions and Binet formulas. Our approach not only extends current research on Cayley–Dickson structures but also provides <i>q</i>-analogues that can effectively address some mathematical problems.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Bi-periodic Fibonacci and Lucas \(2^k\)-ons with q-integer components

  • 期刊论文

  • 发表日期：

    2025年11月4日

查看全文
  • Hafize Gün,
  • Sure Köme

摘要

Quantum calculus, which can be considered a generalization of classical calculus, is essential for many areas of mathematics. Recently, the work on quantum calculus and its applications to mathematics has increased significantly. This article investigates the connection between higher-dimensional algebraic structures and quantum calculus by considering q-biperiodic Fibonacci and q-biperiodic Lucas sequences in Cayley–Dickson algebras. In this article, we examine new relations between Cayley–Dickson algebras, which generalize complex numbers to higher dimensions by systematic doubling, and q-biperiodic Fibonacci and q-biperiodic Lucas sequences, using some useful notations from quantum calculus. We also present algebraic properties of q-biperiodic Fibonacci and q-biperiodic Lucas \(2^k\) -ons, binomial sums, generating functions and Binet formulas. Our approach not only extends current research on Cayley–Dickson structures but also provides q-analogues that can effectively address some mathematical problems.