Gaussian Generalized Olivier Numbers
Yüksel Soykan *
Department of Mathematics, Faculty of Science, Zonguldak Bulent Ecevit University, 67100, Zonguldak, Turkey.
Ikra Celebi
Department of Mathematics, Faculty of Science, Zonguldak Bulent Ecevit University, 67100, Zonguldak, Turkey.
Salih Zeki Tandogan
Department of Mathematics, Faculty of Science, Zonguldak Bulent Ecevit University, 67100, Zonguldak, Turkey.
Esra Cennet Kocak
Department of Mathematics, Faculty of Science, Zonguldak Bulent Ecevit University, 67100, Zonguldak, Turkey.
*Author to whom correspondence should be addressed.
Abstract
This study investigates the structural properties and potential applications of the Gaussian generalized Olivier numbers, contributing to the broader theoretical landscape of integer sequences. Through a combination of algebraic and analytic methods, we establish new recurrence relations, summation identities, and diverse representations for these sequences. In particular, we derive a Binet-type expression, construct generating functions, and develop matrix formulations, while also presenting Simson’s formula as an alternative analytical tool. Special attention is devoted to two notable cases–the Gaussian Olivier numbers and the Gaussian Olivier–Lucas numbers–whose distinctive features highlight the richness of the framework. Our findings demonstrate that these sequences exhibit unique combinatorial behaviors, making them relevant for applications such as coding theory.
Keywords: Olivier numbers, Gaussian Olivier numbers, Pell-Padovan numbers, Pell-Perrin numbers