Computing Hyperfibonacci Numbers by Means of Matrix Transformations and Jordan Forms

Mathematics > Number Theory [Submitted on 16 Jun 2026] Title:Computing Hyperfibonacci Numbers by Means of Matrix Transformations and Jordan Forms View PDF HTML (experimental)Abstract:The Hyperfibonacci sequence of the $r$th generation is defined recursively as a generalization of Fibonacci numbers, where each term is obtained by summing the terms of the Hyperfibonacci sequence of the preceding generation. We introduce the transformation matrix for Hyperfibonacci numbers, which enables us to determine the next term in a given generation. We explore the algebraic structure of that matrix, and its power of $n$, similarity transformations between these matrices and their Jordan canonical forms. Finally, we analyze the powers of these matrices using their Jordan forms, obtaining compact and elegant formulas for expressing $r$-generation Hyperfibonacci numbers in terms of Fibonacci numbers. Submission history From: Nevena Jakovcevic Stor [view email][v1] Tue, 16 Jun 2026 09:39:36 UTC (8 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.