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    An extended framework for bihyperbolic generalized Tribonacci numbers
    (Ankara University, 2024) Gürses, Nurten; İşbilir, Zehra
    The aim of this article is to identify and analyze a new type special number system which is called bihyperbolic generalized Tribonacci numbers (BGTN for short). For this purpose, we give both classical and several new properties such as; recurrence relation, Binet formula, generating function, exponential generating function, summation formulae, matrix formula, and special determinant equations of BGTN . Also, the system of BGTN is quite a big family and includes several type special cases with respect to initial values and $r,~ s, ~t$ values, we give the subfamilies and special cases of it. In addition to these, we construct some numerical algorithms including recurrence relation and special two types determinant equations related to calculating the terms of this new type special number system. Then, we examine several properties by taking two special cases and including some illustrative numerical examples.
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    On the combined Jacobsthal-Padovan generalized quaternions
    (Tbilisi Centre Math Sci, 2022) Gürses, Nurten; İşbilir, Zehra
    In this article, we examine the combined Jacobsthal-Padovan (CJP) generalized quaternions with four special cases: Jacobsthal-Padovan, Jacobsthal-Perrin, adjusted Jacobsthal-Padovan and modified Jacobsthal-Padovan generalized quaternions. Then, recurrence relation, generating function, Binet-like formula and exponential generating function of these quaternions are examined. In addition to this, some new properties, special determinant equations, matrix formulas and summation formulas are discussed.
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    Padovan, Perrin and Pell-Padovan Dual Quaternions
    (2023) İşbilir, Zehra; Gürses, Nurten
    In this present study, we intend to determine the Padovan, Perrin and Pell-Padovan dual quaternions with nonnegative and negative subscripts. In line with this purpose, we construct some new properties such as; special determinant equalities, new recurrence relations, matrix formulas, Binet-like formulas, generating functions, exponential generating functions, summation formulas, and binomial properties for these special dual quaternions.