Matrix representations of an extended family of Horadam polynomials

Hayrullah Özimamoğlu
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 214–228
DOI: 10.7546/nntdm.2026.32.1.214-228
Full paper (PDF, 291 Kb)

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Authors and affiliations

Hayrullah Özimamoğlu
Departments of Mathematics, Faculty of Arts and Sciences, Nevşehir Hacı Bektaş Veli University
Nevşehir, Türkiye

Abstract

In this article, we describe the concept of -Horadam polynomials, which is a generalization of the classical Horadam polynomials . We derive essential characteristics of these newly defined polynomials, such as their generating function, a Binet-type expression, several combinatorial relations, and summation identities. Subsequently, we construct the infinite matrix of d-Horadam polynomials, which can be represented as a Riordan array. Moreover, by employing the Riordan approach, we establish two distinct decompositions of the infinite Pascal matrix involving the d-Horadam polynomials.

Keywords

• Horadam polynomials
• d-Horadam polynomials
• Pascal matrix
• Riordan matrix

2020 Mathematics Subject Classification

• 11B39
• 11B83
• 05A15
• 05A19
• 15A23

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Manuscript history

• Received: 14 October 2025
• Revised: 20 February 2026
• Accepted: 19 March 2026
• Online First: 20 March 2026

Copyright information

Ⓒ 2026 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).