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    Bertrand partner P-Trajectories in the Euclidean 3-space e3
    (Ankara Univ, Fac Sci, 2023) Isbilir, Zehra; Ozen, Kahraman Esen; Tosun, Murat
    The concept of a pair of curves, called as Bertrand partner curves, was introduced by Bertrand in 1850. Bertrand partner curves have been stud-ied widely in the literature from past to present. In this study, we take into account of the concept of Bertrand partner trajectories according to Positional Adapted Frame (PAF) for the particles moving in 3-dimensional Euclidean space. Some characterizations are given for these trajectories with the aid of the PAF elements. Then, we obtain some special cases of these trajectories. Moreover, we provide a numerical example.
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    Examination of generalized Tribonacci dual quaternions
    (Univ Tartu Press, 2023) Isbilir, Zehra; Guerses, Nurten
    This manuscript deals with introducing and discussing of a new type dual quaternions which are named generalized Tribonacci dual quaternions (GTDQ, for short). For this purpose, several new properties, such as Binet formula, generating function, exponential generating function, matrix formula, and determinant equations, are established. In addition to these, some numerical algorithms are constructed. In the last part, some special cases of the family of the GTDQ are examined regarding r, s, t values and initial values considering concluded results.
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    Generalized Padovan Pauli Quaternions
    (Springer Science and Business Media Deutschland GmbH, 2025) Isbilir, Zehra; Doğan Yazıcı, Bahar; Tosun, Murat
    The main purpose of this study is to construct a new type special number system which is defined as generalized Padovan Pauli quaternion with non-negative and negative subscripts. Furthermore, we give some special cases with respect to the initial values and examine them, as well. We obtain not only new equations but also recurrence relations, Binet formulas, generating functions, exponential generating functions, summation formulas, and special determinant equalities with a numerical example regarding this new number system. After all, we construct algorithms for calculating the generalized Padovan Pauli quaternions with non-negative and negative subscripts. Then, we present the R-linear transformation of this new type special Pauli quaternions. © 2025 Elsevier B.V., All rights reserved.
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    Generalized Rectifying Ruled Surfaces of Special Singular Curves
    (Ovidius Univ Press, 2023) Isbilir, Zehra; Yazici, Bahar Dogan; Tosun, Murat
    In this study, generalized rectifying ruled surfaces of Frenet-type framed base curves in the three-dimensional Euclidean space are introduced. These surfaces are a generalization of not only the tangent and binormal surfaces of Frenet-type framed base curves, but also the tangent and binormal surfaces of regular curves. Additionally, we present some geometric characterizations and properties of these surfaces. Then, the singular point classes of the surface are scrutinized and the conditions for being a cross-cap surface are stated. Moreover, generalized rectifying surfaces are examined as framed surfaces by using the framed surface theory, and we investigate the basic invariants and curvatures of them. Then, several illustrative examples with figures are given to support the theorems and results.
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    Generalized ruled surfaces in Myller configuration
    (University of Nis, 2025) Isbilir, Zehra; Doğan Yazıcı, Bahar; Tosun, Murat
    In this paper, we introduce a quite big ruled surface family, which is called generalized ruled surfaces with Frenet-type frame in Myller configuration for Euclidean 3-space. This paper especially improves the theory of surfaces with respect to ruled surfaces and presents the relationships between the usual theory of curves and the theory of surfaces with Myller configuration. We investigate some special type ruled surfaces, such as rectifying-type ruled surfaces, osculating-type ruled surfaces, tangent-type ruled surfaces and trajectory ruled surfaces with Frenet-type frame in Myller configuration for E3 . We also give some particular cases of these ruled surfaces, as well. Since the geometry of versor fields along a curve with Frenet-type frame in Myller configuration for E3 is a generalization of the usual theory of curves in classical Euclidean space, the surface theory of versor fields along a curve with Frenet-type frame in Myller configuration for E3 is a generalization of the usual theory of surfaces in classical Euclidean space, as well. Then, we establish some numerical examples with some illustrative figures with respect to the ruled surfaces in Myller configuration in order to solidify and concretize the given results. © 2025 Elsevier B.V., All rights reserved.
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    GENERALIZED SMARANDACHE CURVES WITH FRENET-TYPE FRAME
    (Honam Mathematical Soc, 2024) Isbilir, Zehra; Tosun, Murat
    In this study, we investigate Smarandache curves with Frenettype frame in Myller configuration for Euclidean 3-space E 3 . Also, we introduce some characterizations and invariants of them. Then, we construct a numerical example with respect to these special Smarandache curves in order to understand the obtained materials.
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    HORADAM 3- PARAMETER GENERALIZED QUATERNIONS
    (Honam Mathematical Soc, 2024) Isbilir, Zehra; Curses, Nurten
    The purpose of this article is to bring together the Horadam numbers and 3-parameter generalized quaternions, which are a general form of the quaternion algebra according to 3-parameters. With this purpose, we introduce and examine a new type of quite big special numbers system, which is called Horadam 3-parameter generalized quaternions (shortly, Horadam 3PGQs), and special cases of them. Besides, we compute both some new equations and classical well-known equations such as; Binet formulas, generating function, exponential generating function, Poisson generating function, sum formulas, Cassini identity, polar representation, and matrix equation. Furthermore, this article concludes by presenting the determinant, characteristic polynomial, characteristic equation, eigenvalues, and eigenvectors in relation to the matrix representation of Horadam 3PGQ.
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    Mannheim partner trajectories related to pafors
    (Univ Nis, 2024) Isbilir, Zehra; Ozen, Kahraman Esen; Tosun, Murat
    In this study, we consider the concept of Mannheim partner trajectories related to the Positional Adapted Frame on Regular Surfaces (PAFORS) for the particles moving on the different regular surfaces in Euclidean 3 -space. We give the relations between the PAFORS elements of these aforementioned trajectories. Also, we obtain the relations between Darboux basis vectors of these trajectories. Furthermore, some special cases of these trajectories are written.
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    A New Insight on Rectifying-Type Curves in Euclidean 4-Space
    (Int Electronic Journal Geometry, 2023) Isbilir, Zehra; Tosun, Murat
    In this study, our purpose is to determine the generalized rectifying-type curves with Frenet-type frame in Myller configuration for Euclidean 4-space E4. Also, some characterizations of them are given. We construct some correlations between curvatures and invariants of generalized rectifyingtype curves. Additionally, we obtain an illustrative example with respect to the rectifying-type curves with Frenet-type frame in Myller configuration for Euclidean 4-space E4.
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    On generalized osculating-type curves in Myller configuration
    (Ovidius Univ Press, 2024) Isbilir, Zehra; Tosun, Murat
    In this study, we examine osculating-type curves with Frenet-type frame in Myller configuration for Euclidean 3-space E-3. We present the necessary characterizations for a curve to be an osculating-type curve. Characterizations originating from the natural structure of Myller configuration are a generalization of osculating curves according to the Frenet frame. Also, we introduce some new results that are not valid for osculating curves. Then, we give an illustrative numerical example supported by a figure.
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    On special curves in Lie groups with Myller configuration
    (Wiley, 2024) Isbilir, Zehra; Dogan Yazici, Bahar; Tosun, Murat
    In this study, we determine a new type comprehensive frame, which is called the generalized Frenet-type frame in three-dimensional Lie groups with Myller configurations, and it includes several special and classical type frames for Euclidean 3-space and three-dimensional Lie groups. After constructing this new comprehensive frame, we obtain derivative formulas with the help of the Lie curvature. In addition, we define some special type curves. The geometry of versor fields along a curve with Frenet-type frame in three-dimensional Lie groups with Myller configurations is a generalization of the usual theory of curves. Since this particular relationship, the osculating-type and rectifying-type curves with Frenet-type frame in three-dimensional Lie groups with Myller configurations include some special cases for osculating and rectifying curves in different spaces.
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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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    Padovan and Perrin hyperbolic spinors
    (Springer Heidelberg, 2025) Isbilir, Zehra; Kosal, Isil Arda; Tosun, Murat
    In this study, we intend to bring together Padovan and Perrin number sequences, which are one of the most popular third-order recurrence sequences, and hyperbolic spinors, which are used in several disciplines from physics to mathematics, with the help of the split quaternions. This paper especially improves the relationship between hyperbolic spinors, both a physical and mathematical concept, and number theory. For this aim, we combine the hyperbolic spinors and Padovan and Perrin numbers concerning the split Padovan and Perrin quaternions, and we determine two new special recurrence sequences named Padovan and Perrin hyperbolic spinors. Then, we give Binet formulas, generating functions, exponential generating functions, Poisson generating functions, and summation formulas. Additionally, we present some matrix and determinant equations with respect to them. Besides, we construct some special equations that give relations between Padovan and Perrin hyperbolic spinors and Padovan and Perrin numbers. Further, we give a short introduction for (s, t)-Padovan and (s, t)-Perrin hyperbolic spinors in order to shed light on future studies.
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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.
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    Pentanacci and Pentanacci-Lucas hybrid numbers
    (Taru Publications, 2023) Isbilir, Zehra; Gürses, 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. © 2023 Elsevier B.V., All rights reserved.
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    Spinor representations of framed curves in the three-dimensional lie groups
    (Taru Publications, 2023) Isbilir, Zehra; Yazici, Bahar Dogan; Tosun, Murat
    In this study, we determine the spinor representations of special singular curves (framed curves) in the three-dimensional Lie groups with a biinvariant metric. Also, we construct spinor framed equations for some special cases and obtain the relations between the spinor representations of the general frame and adapted frame along the framed curves in the three-dimensional Lie groups. Then, we give some geometric properties and results with respect to them.
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    SPINOR REPRESENTATIONS OF PAFORS IN E3
    (Editura Bibliotheca-Bibliotheca Publ House, 2024) Isbilir, Zehra; Ozen, Kahraman Esen; Guner, Mehmet
    In this paper, we introduce the spinor representations of PAFORS for the trajectories endowed with PAFORS on regular surfaces of Euclidean 3-space E-3. We find the spinor equations of PAFORS vectors. Moreover, we obtain the relations between spinor representations of PAFORS and Darboux frame. Then, we give some geometric interpretations and results concerned with this relationship.
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    Spinor Representations of Positional Adapted Frame in the Euclidean 3-Space
    (Int Electronic Journal Geometry, 2023) Isbilir, Zehra; Ozen, Kahraman Esen; Guner, Mehmet
    The main goal of this paper is to study together the spinors, which have a major place in several disciplines from mathematics to physics, and Positional Adapted Frame (PAF) which is a new frame that attracts the attention of many researchers. In accordance with this purpose, we introduce the spinor representations for the trajectories endowed with PAF in the Euclidean 3-space E3, and construct the spinor equations of PAF vectors. Then, we find the relations between spinor representations of PAF and Serret-Frenet frame. Also we give some results and present some geometric interpretations with respect to this relationship. Moreover, we present an illustrative numerical example in order to support the given theorems and results.