On Binomial Transforms: A Study of All Horadam-Type Sequences

Year 2025, Volume: 13 Issue: 2, 198 - 211, 31.10.2025

Faruk Kaplan , Arzu Özkoç Öztürk

Abstract

This article establishes general recurrence relations, including the binomial transform, for all integer sequences with a second-order recurrence relation commonly studied in the literature. We introduce a new generalization of binomial transforms for Horadam-type sequences and investigate the binomial, $k-$binomial, falling $% k-$binomial, and rising $k-$binomial transforms on these sequences. Additionally, we derive the corresponding recurrence relations, Binet formulas, generating functions and Pascal's triangle for each of the four transforms.

Keywords

Binomial transforms , Horadam sequence , Binet formula.

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