{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T06:17:44Z","timestamp":1771049864383,"version":"3.50.1"},"reference-count":71,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T00:00:00Z","timestamp":1749168000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Since Hamilton proposed quaternions as a system of numbers that does not satisfy the ordinary commutative rule of multiplication, quaternion algebras have played an important role in many mathematical and physical studies. This paper introduces the generalized notion of Pauli Fibonacci polynomial quaternions, a definition that incorporates the advantages of the Fibonacci number system augmented by the Pauli matrix structure. With the concept presented in the study, it aims to provide material that can be used in a more in-depth understanding of the principles of coding theory and quantum physics, which contribute to the confidentiality needed by the digital world, with the help of quaternions. In this study, an approach has been developed by integrating the advantageous and consistent structure of quaternions used to solve the problem of system lock-up and unresponsiveness during rotational movements in robot programming, the mathematically compact and functional form of Pauli matrices, and a generalized version of the Fibonacci sequence, which is an application of aesthetic patterns in nature.<\/jats:p>","DOI":"10.3390\/axioms14060449","type":"journal-article","created":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T11:08:31Z","timestamp":1749208111000},"page":"449","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Generalized Pauli Fibonacci Polynomial Quaternions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4685-7101","authenticated-orcid":false,"given":"Bahad\u0131r","family":"Y\u0131lmaz","sequence":"first","affiliation":[{"name":"Department of Mathematics, Art and Science Faculty, Zonguldak B\u00fclent Ecevit University, 67100 Zonguldak, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6300-6889","authenticated-orcid":false,"given":"Nazmiye","family":"G\u00f6n\u00fcl Bilgin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Art and Science Faculty, Zonguldak B\u00fclent Ecevit University, 67100 Zonguldak, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1895-211X","authenticated-orcid":false,"given":"Y\u00fcksel","family":"Soykan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Art and Science Faculty, Zonguldak B\u00fclent Ecevit University, 67100 Zonguldak, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Nagashima, Y. (2011). Elementary Particle Physics: Quantum Field Theory and Particles, John Wiley & Sons.","DOI":"10.1002\/9783527630097"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Dunlap, R.A. (2018). Particle Physics, Morgan & Claypool Publishers.","DOI":"10.1088\/2053-2571\/aae6d3"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1080\/14786449708621070","article-title":"On the Nature of the Electron","volume":"44","author":"Thomson","year":"1987","journal-title":"Philos. Mag."},{"key":"ref_4","first-page":"692","article-title":"The Discovery of the Neutron","volume":"136","author":"Chadwick","year":"1932","journal-title":"Proc. R. Soc. Lond. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF01351864","article-title":"Versuch einer Theorie der \u03b2-Strahlen","volume":"88","author":"Fermi","year":"1934","journal-title":"Z. Phys."},{"key":"ref_6","first-page":"48","article-title":"On the Interaction of Elementary Particles I","volume":"17","author":"Yukawa","year":"1935","journal-title":"Proc. Phys. Math. Soc. Jpn."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Leader, E. (2001). Spin in Particle Physics, Cambridge University Press.","DOI":"10.1017\/CBO9780511524455"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bandyopadhyay, S., and Cahay, M. (2008). Introduction to Spintronics, CRC Press.","DOI":"10.1201\/9781420004748"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Feynman, R.P., Leighton, R.B., and Sands, M. (1965). The Feynman Lectures on Physics, Addison-Wesley.","DOI":"10.1119\/1.1972241"},{"key":"ref_10","unstructured":"Landau, L.D., and Lifshitz, E.M. (2013). Quantum Mechanics (Non-Relativistic Theory), Pergamon. [3rd ed.]."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Griffiths, D.J. (1995). Introduction to Quantum Mechanics, Prentice Hall.","DOI":"10.1063\/1.2808172"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1007\/BF02746019","article-title":"On the Equations of Motion for Particles with Arbitrary Spin in Non-relativistic Mechanics","volume":"14","author":"Fushchich","year":"1975","journal-title":"Lett. Nuovo Cimento"},{"key":"ref_13","first-page":"529","article-title":"Spin: All Is Not What It Seems","volume":"38","author":"Morrison","year":"2007","journal-title":"Stud. Hist. Philos. Sci. Part B"},{"key":"ref_14","first-page":"1","article-title":"Dark Spinor Models in Gravitation and Cosmology","volume":"2010","author":"Boehmer","year":"2010","journal-title":"J. High Energy Phys."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Vaz, J., and da Rocha, R. (2016). An Introduction to Clifford Algebras and Spinors, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198782926.001.0001"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/BF00674438","article-title":"Covariant, Algebraic, and Operator Spinors","volume":"29","author":"Figueiredo","year":"1990","journal-title":"Int. J. Theor. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"425","DOI":"10.2307\/2371218","article-title":"Spinors in n Dimensions","volume":"57","author":"Brauer","year":"1935","journal-title":"Am. J. Math."},{"key":"ref_18","unstructured":"Cartan, E. (1966). Theory of Spinors, Dover."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Chevalley, C. (1954). The Algebraic Theory of Spinors, Columbia University Press.","DOI":"10.7312\/chev93056"},{"key":"ref_20","first-page":"85","article-title":"Differential Forms as Spinors","volume":"29","author":"Graf","year":"1978","journal-title":"Ann. Inst. Henri Poincar\u00e9"},{"key":"ref_21","unstructured":"Riesz, M. (1958). Clifford Numbers and Spinors, Institute for Fluid Mechanics and Applied Mathematics, University of Maryland. Lecture Notes No. 38."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Hestenes, D., and Sobezyk, G. (1984). Clifford Algebra to Geometrical Calculus, D. Reidel.","DOI":"10.1007\/978-94-009-6292-7"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3533","DOI":"10.1007\/s10773-015-2596-0","article-title":"Spinor Structure and Internal Symmetries","volume":"54","author":"Varlamov","year":"2015","journal-title":"Int. J. Theor. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.1103\/PhysRev.125.1067","article-title":"Symmetries of Baryons and Mesons","volume":"125","year":"1962","journal-title":"Phys. Rev."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/0029-5582(61)90134-1","article-title":"Derivation of Strong Interactions from a Gauge Invariance","volume":"26","year":"1961","journal-title":"Nucl. Phys."},{"key":"ref_26","unstructured":"Wrede, R.C. (2013). Introduction to Vector and Tensor Analysis, Courier Corporation."},{"key":"ref_27","unstructured":"Corben, H.C., and Stehle, P. (1960). Classical Mechanics, John Wiley & Sons. [2nd ed.]."},{"key":"ref_28","unstructured":"Montague, B.W. (1981). An Introduction to Spinor Algebra for Polarized Beams, CERN. CLRN\/ISR-TH\/80-02."},{"key":"ref_29","first-page":"424","article-title":"On a New Species of Imaginary Quantities Connected with a Theory of Quaternions","volume":"2","author":"Hamilton","year":"1843","journal-title":"Proc. R. Ir. Acad."},{"key":"ref_30","unstructured":"Tait, P.G., and Kelland, P. (2024). Introduction to Quaternions with Numerous Examples, Macmillan and Co., Ltd."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Herath, D., and St-Onge, D. (2022). Foundations of Robotics: A Multidisciplinary Approach with Python and ROS, Springer Nature.","DOI":"10.1007\/978-981-19-1983-1"},{"key":"ref_32","first-page":"267","article-title":"Representing Rotation in Simulink Using Quaternion","volume":"8","author":"Perumal","year":"2014","journal-title":"Appl. Math."},{"key":"ref_33","unstructured":"Eric, M.J., and Paul, F. (2025, May 05). Apollo Lunar Surface Journal: Gimbal Angles, Gimbal Lock, and a Fourth Gimbal for Christmas, Available online: http:\/\/www.hq.nasa.gov\/alsj\/gimbals.html."},{"key":"ref_34","unstructured":"Hladik, J. (1999). Spinors in Physics, Springer Science & Business Media."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"610","DOI":"10.1098\/rspa.1928.0023","article-title":"The Quantum Theory of the Electron","volume":"117","author":"Dirac","year":"1928","journal-title":"Proc. R. Soc. Lond. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1007\/BF01397326","article-title":"Zur Quantenmechanik des magnetischen Elektrons","volume":"43","author":"Pauli","year":"1927","journal-title":"Z. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"90","DOI":"10.3367\/UFNe.2021.06.039000","article-title":"Discovery of the Relativistic Schr\u00f6dinger Equation","volume":"65","author":"Barley","year":"2022","journal-title":"Phys. Uspekhi"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF01236599","article-title":"Development of Spinor Descriptions of Rotational Mechanics from Euler\u2019s Rigid Body Displacement Theorem","volume":"32","author":"Vivarelli","year":"1984","journal-title":"Celest. Mech."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1007\/s10701-011-9566-z","article-title":"Division Algebras and Quantum Theory","volume":"42","author":"Baez","year":"2012","journal-title":"Found. Phys."},{"key":"ref_40","first-page":"15","article-title":"Generalized Quaternions and Their Algebraic Properties","volume":"64","author":"Jafari","year":"2015","journal-title":"Commun. Fac. Sci. Univ. Ank. Ser. A1"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40590-021-00386-4","article-title":"Generalized Commutative Quaternions of the Fibonacci Type","volume":"28","year":"2022","journal-title":"Bol. Soc. Mat. Mex."},{"key":"ref_42","first-page":"145","article-title":"A Representation of de Moivre\u2019s Formula over Pauli-Quaternions","volume":"9","author":"Kim","year":"2017","journal-title":"Fibonacci Q."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1016\/0031-8914(66)90050-4","article-title":"The Properties of the Pauli Matrices A, B, C and the Conjugation of Charge","volume":"32","author":"Longe","year":"1966","journal-title":"Physica"},{"key":"ref_44","unstructured":"Arfken, G. (1985). Mathematical Methods for Physicists, Academic Press."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1080\/00150517.1969.12431146","article-title":"A Note on Fibonacci Quaternions","volume":"7","author":"Iyer","year":"1969","journal-title":"Fibonacci Q."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1080\/00029890.1961.11989696","article-title":"A Generalized Fibonacci Sequence","volume":"68","author":"Horadam","year":"1961","journal-title":"Am. Math. Mon."},{"key":"ref_47","first-page":"23","article-title":"On Generalized Fibonacci Polynomials: Horadam Polynomials","volume":"11","author":"Soykan","year":"2023","journal-title":"Earthline J. Math. Sci."},{"key":"ref_48","first-page":"1","article-title":"On Generalized (r, s)-numbers","volume":"8","author":"Soykan","year":"2020","journal-title":"Int. J. Adv. Appl. Math. Mech."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s00006-011-0317-1","article-title":"On Fibonacci Quaternions","volume":"22","author":"Halici","year":"2012","journal-title":"Adv. Appl. Clifford Algebras"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"289","DOI":"10.2307\/2313129","article-title":"Complex Fibonacci Numbers and Fibonacci Quaternions","volume":"70","author":"Horadam","year":"1963","journal-title":"Am. Math. Mon."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1007\/s41478-020-00295-1","article-title":"On Fibonacci quaternions","volume":"29","author":"Kizilates","year":"2021","journal-title":"J. Anal."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"\u00c7olak, E.G., G\u00f6n\u00fcl Bilgin, N., and Soykan, Y. (2024). On Generalized Fibospinomials: Generalized Fibonacci Polynomial Spinors. Symmetry, 16.","DOI":"10.20944\/preprints202402.0910.v1"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1007\/s00006-014-0458-0","article-title":"Fibonacci Generalized Quaternions","volume":"24","author":"Akyigit","year":"2014","journal-title":"Adv. Appl. Clifford Algebras"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.chaos.2015.10.021","article-title":"On a New Generalization of Fibonacci Quaternions","volume":"82","author":"Tan","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1007\/s00006-015-0626-x","article-title":"A Generalization of Fibonacci and Lucas Quaternions","volume":"26","author":"Polatli","year":"2016","journal-title":"Adv. Appl. Clifford Algebras"},{"key":"ref_56","unstructured":"Halici, S. (2016). On a Generalization for Quaternion Sequences. arXiv."},{"key":"ref_57","first-page":"707","article-title":"Some Properties and Identities of Hyperbolic Generalized k-Horadam Quaternions and Octonions","volume":"9","author":"Prasad","year":"2024","journal-title":"Commun. Combin. Optim."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Y\u0131lmaz, F., and \u00d6zkan, M. (2022). On the Generalized Gaussian Fibonacci Numbers and Horadam Hybrid Numbers: A Unified Approach. Axioms, 11.","DOI":"10.3390\/axioms11060255"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"184","DOI":"10.7546\/nntdm.2021.27.3.184-193","article-title":"Pauli\u2013Fibonacci Quaternions","volume":"27","year":"2021","journal-title":"Notes Number Theory Discret. Math."},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Azak, A.Z. (2022). Pauli Gaussian Fibonacci and Pauli Gaussian Lucas Quaternions. Mathematics, 10.","DOI":"10.3390\/math10244655"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1","DOI":"10.7546\/nntdm.2023.29.1.1-16","article-title":"Pauli\u2013Leonardo Quaternions","volume":"29","author":"Tosun","year":"2023","journal-title":"Notes Number Theory Discret. Math."},{"key":"ref_62","unstructured":"\u0130sbilir, Z., Yaz\u0131c\u0131, B.D., and Tosun, M. (2025, June 03). Incomplete Generalized (p, q, r)-Tribonacci Pauli Quaternion Polynomials. Available online: https:\/\/www.researchgate.net\/profile\/Zehra-Isbilir\/publication\/373019532."},{"key":"ref_63","first-page":"23","article-title":"Quaternion Recurrence Relations","volume":"2","author":"Horadam","year":"1993","journal-title":"Ulam Q."},{"key":"ref_64","first-page":"8","article-title":"Encryption of Data Streams Using Pauli Spin 12 Matrices and Finite State Machine","volume":"37","author":"Gandhi","year":"2012","journal-title":"Int. J. Comput. Appl."},{"key":"ref_65","first-page":"37","article-title":"Secure Cryptosystem Based on Braiding\/Entanglement of Pauli 3\/2 Matrices","volume":"143","author":"Kumar","year":"2016","journal-title":"Int. J. Comput. Appl."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/s11128-020-02728-z","article-title":"A Unitary Operator Construction Solution Based on Pauli Group for Maximal Dense Coding with a Class of Symmetric States","volume":"19","author":"Liu","year":"2020","journal-title":"Quantum Inf. Process."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"469124","DOI":"10.1155\/2010\/469124","article-title":"Fast Constructions of Quantum Codes Based on Residues Pauli Block Matrices","volume":"2010","author":"Guo","year":"2010","journal-title":"Adv. Math. Phys."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"866","DOI":"10.22331\/q-2022-12-01-866","article-title":"Quantum Fully Homomorphic Encryption by Integrating Pauli One-Time Pad with Quaternions","volume":"6","author":"Ma","year":"2022","journal-title":"Quantum"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"43","DOI":"10.22331\/q-2018-01-04-43","article-title":"Fault-Tolerant Quantum Computing in the Pauli or Clifford Frame with Slow Error Diagnostics","volume":"2","author":"Chamberland","year":"2018","journal-title":"Quantum"},{"key":"ref_70","first-page":"277","article-title":"k-Order Fibonacci Polynomials on AES-Like Cryptology","volume":"131","author":"Asci","year":"2022","journal-title":"CMES\u2013Comput. Model. Eng. Sci."},{"key":"ref_71","unstructured":"Abraham, S., and Bhagwat, A.A. (2024). PT-Symmetry in 2 \u00d7 2 Matrix Polynomials Formed by Pauli Matrices. arXiv."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/449\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:47:54Z","timestamp":1760032074000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/449"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,6]]},"references-count":71,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["axioms14060449"],"URL":"https:\/\/doi.org\/10.3390\/axioms14060449","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,6]]}}}