TY - JOUR T1 - Properties of Gaussian Generalized Leonardo Numbers TT - Gaussian Genelleştirilmiş Leonardo Sayılarının Özellikleri AU - Dikmen, Can Murat PY - 2025 DA - April Y2 - 2025 DO - 10.7212/karaelmasfen.1578154 JF - Karaelmas Fen ve Mühendislik Dergisi PB - Zonguldak Bülent Ecevit Üniversitesi WT - DergiPark SN - 2146-7277 SP - 134 EP - 145 VL - 15 IS - 1 LA - en AB - In this research, we introduce and thoroughly examine Gaussian generalized Leonardo numbers, focusing on three distinct cases: Gaussian modified Leonardo numbers, Gaussian Leonardo‐Lucas numbers, and Gaussian Leonardo numbers. Our aim is to offer a comprehensive understanding of the behaviour and properties of these sequences. To this end, we perform a detailed analysis, deriving various identities and matrices associated with these sequences. We also explore key mathematical tools such as recurrence relations, Binet’s formulas, generating functions, Simpson’s formula, Honsberger’s identity, and several summation formulas. This multifaceted approach provides valuable insights into the structure and behaviour of these Gaussian-based sequences. The results we present not only extend existing knowledge but also open the door for future studies that could explore further generalizations and applications of Gaussian generalized Leonardo numbers. KW - Gaussian generalized Leonardo numbers KW - Gaussian Leonardo‐Lucas numbers KW - Gaussian modified Leonardo numbers KW - Gaussian Leonardo numbers. N2 - Bu araştırmada, Gaussian genelleştirilmiş Leonardo sayılarını tanıtıyor ve kapsamlı bir şekilde inceliyoruz ve üç farklı duruma odaklanıyoruz: Gaussian modifiye Leonardo sayıları, Gaussian Leonardo-Lucas sayıları ve Gaussian Leonardo sayıları. Amacımız, bu dizilerin davranışı ve özellikleri hakkında kapsamlı bir anlayış sunmaktır.Bu amaçla, bu dizilerle ilişkili çeşitli özdeşlikler ve matrisler türeterek derinlemesine bir analiz gerçekleştiriyoruz. Ayrıca, yineleme bağıntıları, Binet formülleri, üreteç fonksiyonlar, Simpson formülü, Honsberger özdeşliği ve çeşitli toplam formülleri gibi temel matematiksel araçları da araştırıyoruz. Bu çok yönlü yaklaşım, bu Gaussian tabanlı dizilerin yapısı ve davranışı hakkında değerli içgörüler sağlar. Sunduğumuz sonuçlar yalnızca mevcut bilgiyi genişletmekle kalmıyor, aynı zamanda Gaussian genelleştirilmiş Leonardo sayılarının daha fazla genelleştirilmesini ve uygulamasını araştırabilecek gelecekteki çalışmalar için de kapı açıyor CR - Alp, Y., Kocer, EG. 2021. Some properties of Leonardo numbers. Konuralp J. Math; 9 (1): 183–189. CR - Aşçı, M., Gürel, E. 2013. Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers. Ars Combinatoria, 111:53-62. 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