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    On the 3-parameter generalized quaternions with generalized tribonacci numbers components
    (Univ Nis, Fac Sci Math, 2025) Isbilir, Zehra; Gurses, Nurten; Tosun, Murat
    In this paper, we aim to combine 3-parameter generalized quaternions (shortly 3PGQs), which are a general form of the quaternion algebra according to 3-parameters, and generalized Tribonacci number (shortly GTNs), which are also quite a big special number family for third-order recurrence sequences and most general form of all of the third-order recurrence sequences. Namely, we investigate a special new number system called 3-parameter generalized quaternions with generalized Tribonacci numbers components (shortly 3PGQs with GTN components) with both nonnegative and negative subscripts and examine some special cases of them. Then, we construct a Maple code of this special number family. Moreover, we obtain some new and classical well-known equations such as; Binet formulas, generating function, exponential generating function, Poisson generating function, summation formulas, polar representation, and matrix equation. In addition to these, we give also determinant, characteristic polynomial, characteristic equation, eigenvalues, and eigenvectors concerning the matrix representation of 3PGQs with GTN components.
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    Padovan and Perrin generalized quaternions
    (Wiley, 2021) Isbilir, Zehra; Gurses, Nurten
    In this study, we investigate the Padovan (or Cordonnier) and Perrin generalized quaternions. We obtain the new identities for these special quaternions related to matrix forms. We also introduce Binet-like formulae, generating functions, several summation, and binomial properties concerning these quaternions.
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    Pell-Padovan generalized quaternions
    (Bulgarian Acad Science, 2021) Isbilir, Zehra; Gurses, Nurten
    The aim of this article is to introduce Pell-Padovan generalized quaternions. It also derives new properties associated with these and takes into account negative indices. Additionally, it presents generating function, Binet-like formula, Simson formula, matrix representations, and several summation properties.
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    Pentanacci and Pentanacci-Lucas hybrid numbers
    (Taylor & Francis Ltd, 2021) Isbilir, Zehra; Gurses, Nurten
    The aim of this study is to introduce the Pentanacci and Pentanacci-Lucas hybrid numbers. Then, some properties with respect to these special numbers and relations between them are obtained. In addition, matrix formulations, generating functions, Binet-like formulas and some summation formulas of them are examined.