Research Article
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On the Leonardo quaternions sequence

Year 2024, Volume: 53 Issue: 4, 1001 - 1023, 27.08.2024

Patrícia Beites , Paula Maria Machado Cruz Catarino

https://doi.org/10.15672/hujms.1197693
Cited By: 1

Abstract

In the present work, a new sequence of quaternions related to the Leonardo numbers -- named the Leonardo quaternions sequence -- is defined and studied. Binet's formula and certain sum and binomial-sum identities, some of which derived from the mentioned formula, are established. Tagiuri-Vajda's identity and, as consequences, Catalan's identity, d'Ocagne's identity and Cassini's identity are presented. Furthermore, applying Catalan's identity, and the connection between composition algebras and vector cross product algebras, Gelin-Cesàro's identity is also stated and proved. Finally, the generating function, the exponential generating function and the Poisson generating function are deduced. In addition to the results on Leonardo quaternions, known results on Leonardo numbers and on Fibonacci quaternions are extended.

Keywords

Leonardo numbers Leonardo quaternions Binet’s formula (Tagiuri-Vajda’s Sum and Binomial-sum) identities generating function

Supporting Institution

University of Beira Interior, Portugal; University of Trás-os-Montes e Alto Douro, Portugal

Project Number

UIDB/00212/2020, MTM2017-83506-C2-2-P, UIDB/00013/2020, UIDP/00013/2020, UID/CED/00194/2020

Thanks

P. D. Beites was supported by FCT (Fundação para a Ciência e a Tecnologia, Portugal), project UIDB/00212/2020 of CMA-UBI (Centro de Matemática e Aplicações da Universidade da Beira Interior, Portugal), and by project MTM2017-83506-C2-2-P (Spain). The author P. Catarino was supported by FCT, projects UIDB/00013/2020, UIDP/00013/2020 and UID/CED/00194/2020.

References

  • [1] B. Aloui and A. Boussayoud, Generating functions of the product of the k-Fibonacci and k-Pell numbers and Chebyshev polynomials of the third and fourth kind, Math. Eng. Sci. Aerosp. 12, 245-257, 2021.
  • [2] F. Alves and R. Vieira, The Newton fractal’s Leonardo sequence study with the Google Colab, Int. Electron. J. Math. Educ. 15, em0575, 2020.
  • [3] M. Akyiğit, H.H. Kösal and M. Tosun, Split Fibonacci quaternions, Adv. Appl. Clifford Algebras 23, 535-545, 2013.
  • [4] M. Akyiğit, H.H. Kösal and M. Tosun, Fibonacci generalized quaternions, Adv. Appl. Clifford Algebras 24, 631-641, 2014.
  • [5] Y. Alp and E.G. Koçer, Some properties of Leonardo numbers, Konuralp J. Math. 9, 183-189, 2021.
  • [6] P.D. Beites and A.P. Nicolás, An associative triple system of the second kind, Commun. Algebra 44, 5027-5043, 2016.
  • [7] P.D. Beites and A.P. Nicolás, A note on standard composition algebras of types II and III, Adv. Appl. Clifford Algebras 27, 955-964, 2017.
  • [8] P.D. Beites, A.P. Nicolás, P. Saraiva and J. Vitória, Vector cross product differential and difference equations in $\mathbb{R}^3$and in$\mathbb{R}^7$, Electron. J. Linear Algebra 34, 675-686, 2018.
  • [9] G. Bilgici and P. Catarino, Unrestricted pell and pell-lucas quaternions, Int. J. Math. Syst. Sci. 1, Article 816, 2018.
  • [10] G. Bilgici, U. Tokeşer and Z. Ünal, k-Fibonacci and k-Lucas generalized quaternions, Konuralp J. Math. 5, 102-113, 2017.
  • [11] T.R. Blackman and S. Lemurell, Spectral correspondences for Maass waveforms on quaternion groups, J. Number Theory 158, 1-22, 2016.
  • [12] I. Cação, H.R. Malonek and G. Tomaz, Shifted Generalized Pascal Matrices in the Context of Clifford Algebra-Valued Polynomial Sequences, in: Computational Science and its Applications, Lecture Notes in Computer Science 10405, Springer, 2017.
  • [13] N.D. Cahill, J.R. D’Errico, D.A. Narayan and J.Y. Narayan, Fibonacci determinants, Coll. Math. J. 33, 221-225, 2002.
  • [14] P. Catarino, The modified Pell and the modified k-Pell quaternions and octonions, Adv. Appl. Clifford Algebras 26, 577-590, 2016.
  • [15] P. Catarino and A. Borges, On Leonardo numbers, Acta Math. Univ. Comen. 89, 75-86, 2020.
  • [16] P. Catarino and A. Borges, A note on incomplete Leonardo numbers, Integers 20, article A43, 2020.
  • [17] P. Catarino and H. Campos, Incomplete k-Pell, k-Pell-Lucas and modified k-Pell numbers, Hacet. J. Math. Stat. 46, 361-372, 2017.
  • [18] P. Catarino and R. de Almeida, On a quaternionic sequence with Vietoris’ numbers, Filomat 35, 1065-1086, 2021.
  • [19] P. Catarino and R. de Almeida, A note on Vietoris’ number sequence, Mediterr. J. Math. 19, 1-19, 2022.
  • [20] Y. Choo, A generalized quaternion with generalized Fibonacci number components, Appl. Math. Sci. 14, 31-38, 2020.
  • [21] C.B. Çimen and A. İpek, On Pell quaternions and Pell-Lucas quaternions, Adv. Appl. Clifford Algebras 26, 39-51, 2016.
  • [22] J.H. Conway and D.A. Smith, On Quaternions and Octonions: their Geometry, Arithmetic and Symmetry, A K Peters/CRC Press, New York, 2003.
  • [23] N. Correia and R. Pacheco, Harmonic maps of finite uniton number and their canonical elements, Ann. Glob. Anal. Geom. 47, 335-358, 2015.
  • [24] A. Daşdemir, Gelin-Cesàro identities for Fibonacci and Lucas quaternions, Ann. Univ. Paedagog. Crac. Stud. Math. XVIII, 137-144, 2019.
  • [25] E.W. Dijkstra, Smoothsort, an alternative for sorting in situ, Sci. Comput. Program. 1, 223-233, 1982.
  • [26] G.B. Djordjević and H.M. Srivastava, Some generalizations of certain sequences associated with the Fibonacci numbers, J. Indones. Math. Soc. 12, 99-112, 2006.
  • [27] M.I. Falcão, F. Miranda, R. Severino and M.J. Soares, Evaluation schemes in the ring of quaternionic polynomials, BIT Numer. Math. 58, 51-72, 2018.
  • [28] N. Jacobson, Composition algebras and their automorphisms, Rend. Circ. Mat. Palermo 7, 55-80, 1958.
  • [29] S. Halici, On Fibonacci quaternions, Adv. Appl. Clifford Algebras 22, 321-327, 2012.
  • [30] S. Halici and G. Cerda-Morales, On Quaternion-Gaussian Fibonacci numbers and their properties, An. St. Univ. Ovidius Constanta, Ser. Mat. 29, 71-82, 2021.
  • [31] S. Halici and A. Karataş, On a generalization for Fibonacci quaternions, Chaos Soliton Fract. 98, 178-182, 2017.
  • [32] A.F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Am. Math. Mon. 70, 289-291, 1963.
  • [33] A. İpek, On (p, q)-Fibonacci quaternions and their Binet formulas, generating functions and certain binomial sums, Adv. Appl. Clifford Algebrass 27, 1343-1351, 2017.
  • [34] E. Kilic, D. Tasci and P. Haukkanen, On the generalized Lucas sequences by Hessenberg matrices, Ars Combinatoria 95, 383-395, 2010.
  • [35] D. Knuth, The art of computer programming, Addison Wesley Longman, 1997.
  • [36] T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2018.
  • [37] F.S. Leite, The geometry of hypercomplex matrices, Linear Multilinear Algebra 34, 123-132, 1993.
  • [38] J. Morais and I. Cação, Quaternion Zernike spherical polynomials, Math. Comput. 84, 1317-1337, 2015.
  • [39] B.K. Patel and P.K. Ray, On the properties of $(p, q)$-Fibonacci and $(p, q)$-Lucas quaternions, Mathematical Reports 21, 15-25, 2019.
  • [40] E. Polatlı and S. Kesim, A note on Catalan’s identity for the k-Fibonacci quaternions, J. Integer Seq. 18, article 15.8.2, 2015.
  • [41] E. Polatlı, C. Kizilates and S. Kesim, On split k-Fibonacci and k-Lucas quaternions, Adv. Appl. Clifford Algebras 26, 353-362, 2016.
  • [42] E. Polatlı and Y. Soykan, On Generalized Third-order Jacobsthal numbers, Asian Res. J. Math. 17, 1-19, 2021.
  • [43] J.L. Ramírez, Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions, An. St. Univ. Ovidius Constanta, Ser. Mat. 23, 201-212, 2015.
  • [44] N. Saba, A. Boussayoud and K. V. Kanuri, Mersenne Lucas numbers and complete homogeneous symmetric functions, J. Math. Comput. Sci. 24, 127-139, 2022.
  • [45] R. Serôdio, P.D. Beites and J. Vitória, Intersection of a double cone and a line in the split-quaternions context, Adv. Appl. Clifford Algebras 27, 2795-2803, 2017.
  • [46] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, The OEIS Foundation, https://oeis.org, 2021.
  • [47] A. Szynal-Liana and I. Włoch, The Pell quaternions and the Pell octonions, Adv. Appl. Clifford Algebras 26, 435-440, 2016.
  • [48] A. Szynal-Liana and I. Włoch, A note on Jacobsthal quaternions, Adv. Appl. Clifford Algebras 26, 441-447, 2016.
  • [49] E. Tan and H.-H. Leung, Some results on Horadam quaternions, Chaos Soliton Fract. 138, article 109961, 2020.
  • [50] D. Tasci, On k-Jacobsthal and k-Jacobsthal-Lucas quaternions, Journal of Science and Arts 3, 469-476, 2017.
  • [51] U. Tokeşer, Z. Ünal and G. Bilgici, Split Pell and Pell-Lucas quaternions, Adv. Appl. Clifford Algebras 27, 1881-1893, 2017.
  • [52] S. Vajda, Fibonacci & Lucas numbers, and the golden section, Ellis Horwood Ltd., Chichester, England, 1989.
  • [53] R. Vieira, F. Alves and P. Catarino, Relações bidimensionais e identidades da sequência de Leonardo, Revista Sergipana de Matemática e Educação Matemática 4, 156-173, 2019.
  • [54] R. Vieira, M. Mangueira, F. Alves and P. Catarino, A forma matricial dos números de Leonardo, Ciência e Natura 42, article e100, 2020.
  • [55] T. Yağmur, Split Jacobsthal and Jacobsthal-Lucas quaternions, Commun. Math. Appl. 10, 429-438, 2019.

Year 2024, Volume: 53 Issue: 4, 1001 - 1023, 27.08.2024

Patrícia Beites , Paula Maria Machado Cruz Catarino

https://doi.org/10.15672/hujms.1197693
Cited By: 1

Abstract

Project Number

UIDB/00212/2020, MTM2017-83506-C2-2-P, UIDB/00013/2020, UIDP/00013/2020, UID/CED/00194/2020

References

  • [1] B. Aloui and A. Boussayoud, Generating functions of the product of the k-Fibonacci and k-Pell numbers and Chebyshev polynomials of the third and fourth kind, Math. Eng. Sci. Aerosp. 12, 245-257, 2021.
  • [2] F. Alves and R. Vieira, The Newton fractal’s Leonardo sequence study with the Google Colab, Int. Electron. J. Math. Educ. 15, em0575, 2020.
  • [3] M. Akyiğit, H.H. Kösal and M. Tosun, Split Fibonacci quaternions, Adv. Appl. Clifford Algebras 23, 535-545, 2013.
  • [4] M. Akyiğit, H.H. Kösal and M. Tosun, Fibonacci generalized quaternions, Adv. Appl. Clifford Algebras 24, 631-641, 2014.
  • [5] Y. Alp and E.G. Koçer, Some properties of Leonardo numbers, Konuralp J. Math. 9, 183-189, 2021.
  • [6] P.D. Beites and A.P. Nicolás, An associative triple system of the second kind, Commun. Algebra 44, 5027-5043, 2016.
  • [7] P.D. Beites and A.P. Nicolás, A note on standard composition algebras of types II and III, Adv. Appl. Clifford Algebras 27, 955-964, 2017.
  • [8] P.D. Beites, A.P. Nicolás, P. Saraiva and J. Vitória, Vector cross product differential and difference equations in $\mathbb{R}^3$and in$\mathbb{R}^7$, Electron. J. Linear Algebra 34, 675-686, 2018.
  • [9] G. Bilgici and P. Catarino, Unrestricted pell and pell-lucas quaternions, Int. J. Math. Syst. Sci. 1, Article 816, 2018.
  • [10] G. Bilgici, U. Tokeşer and Z. Ünal, k-Fibonacci and k-Lucas generalized quaternions, Konuralp J. Math. 5, 102-113, 2017.
  • [11] T.R. Blackman and S. Lemurell, Spectral correspondences for Maass waveforms on quaternion groups, J. Number Theory 158, 1-22, 2016.
  • [12] I. Cação, H.R. Malonek and G. Tomaz, Shifted Generalized Pascal Matrices in the Context of Clifford Algebra-Valued Polynomial Sequences, in: Computational Science and its Applications, Lecture Notes in Computer Science 10405, Springer, 2017.
  • [13] N.D. Cahill, J.R. D’Errico, D.A. Narayan and J.Y. Narayan, Fibonacci determinants, Coll. Math. J. 33, 221-225, 2002.
  • [14] P. Catarino, The modified Pell and the modified k-Pell quaternions and octonions, Adv. Appl. Clifford Algebras 26, 577-590, 2016.
  • [15] P. Catarino and A. Borges, On Leonardo numbers, Acta Math. Univ. Comen. 89, 75-86, 2020.
  • [16] P. Catarino and A. Borges, A note on incomplete Leonardo numbers, Integers 20, article A43, 2020.
  • [17] P. Catarino and H. Campos, Incomplete k-Pell, k-Pell-Lucas and modified k-Pell numbers, Hacet. J. Math. Stat. 46, 361-372, 2017.
  • [18] P. Catarino and R. de Almeida, On a quaternionic sequence with Vietoris’ numbers, Filomat 35, 1065-1086, 2021.
  • [19] P. Catarino and R. de Almeida, A note on Vietoris’ number sequence, Mediterr. J. Math. 19, 1-19, 2022.
  • [20] Y. Choo, A generalized quaternion with generalized Fibonacci number components, Appl. Math. Sci. 14, 31-38, 2020.
  • [21] C.B. Çimen and A. İpek, On Pell quaternions and Pell-Lucas quaternions, Adv. Appl. Clifford Algebras 26, 39-51, 2016.
  • [22] J.H. Conway and D.A. Smith, On Quaternions and Octonions: their Geometry, Arithmetic and Symmetry, A K Peters/CRC Press, New York, 2003.
  • [23] N. Correia and R. Pacheco, Harmonic maps of finite uniton number and their canonical elements, Ann. Glob. Anal. Geom. 47, 335-358, 2015.
  • [24] A. Daşdemir, Gelin-Cesàro identities for Fibonacci and Lucas quaternions, Ann. Univ. Paedagog. Crac. Stud. Math. XVIII, 137-144, 2019.
  • [25] E.W. Dijkstra, Smoothsort, an alternative for sorting in situ, Sci. Comput. Program. 1, 223-233, 1982.
  • [26] G.B. Djordjević and H.M. Srivastava, Some generalizations of certain sequences associated with the Fibonacci numbers, J. Indones. Math. Soc. 12, 99-112, 2006.
  • [27] M.I. Falcão, F. Miranda, R. Severino and M.J. Soares, Evaluation schemes in the ring of quaternionic polynomials, BIT Numer. Math. 58, 51-72, 2018.
  • [28] N. Jacobson, Composition algebras and their automorphisms, Rend. Circ. Mat. Palermo 7, 55-80, 1958.
  • [29] S. Halici, On Fibonacci quaternions, Adv. Appl. Clifford Algebras 22, 321-327, 2012.
  • [30] S. Halici and G. Cerda-Morales, On Quaternion-Gaussian Fibonacci numbers and their properties, An. St. Univ. Ovidius Constanta, Ser. Mat. 29, 71-82, 2021.
  • [31] S. Halici and A. Karataş, On a generalization for Fibonacci quaternions, Chaos Soliton Fract. 98, 178-182, 2017.
  • [32] A.F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Am. Math. Mon. 70, 289-291, 1963.
  • [33] A. İpek, On (p, q)-Fibonacci quaternions and their Binet formulas, generating functions and certain binomial sums, Adv. Appl. Clifford Algebrass 27, 1343-1351, 2017.
  • [34] E. Kilic, D. Tasci and P. Haukkanen, On the generalized Lucas sequences by Hessenberg matrices, Ars Combinatoria 95, 383-395, 2010.
  • [35] D. Knuth, The art of computer programming, Addison Wesley Longman, 1997.
  • [36] T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2018.
  • [37] F.S. Leite, The geometry of hypercomplex matrices, Linear Multilinear Algebra 34, 123-132, 1993.
  • [38] J. Morais and I. Cação, Quaternion Zernike spherical polynomials, Math. Comput. 84, 1317-1337, 2015.
  • [39] B.K. Patel and P.K. Ray, On the properties of $(p, q)$-Fibonacci and $(p, q)$-Lucas quaternions, Mathematical Reports 21, 15-25, 2019.
  • [40] E. Polatlı and S. Kesim, A note on Catalan’s identity for the k-Fibonacci quaternions, J. Integer Seq. 18, article 15.8.2, 2015.
  • [41] E. Polatlı, C. Kizilates and S. Kesim, On split k-Fibonacci and k-Lucas quaternions, Adv. Appl. Clifford Algebras 26, 353-362, 2016.
  • [42] E. Polatlı and Y. Soykan, On Generalized Third-order Jacobsthal numbers, Asian Res. J. Math. 17, 1-19, 2021.
  • [43] J.L. Ramírez, Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions, An. St. Univ. Ovidius Constanta, Ser. Mat. 23, 201-212, 2015.
  • [44] N. Saba, A. Boussayoud and K. V. Kanuri, Mersenne Lucas numbers and complete homogeneous symmetric functions, J. Math. Comput. Sci. 24, 127-139, 2022.
  • [45] R. Serôdio, P.D. Beites and J. Vitória, Intersection of a double cone and a line in the split-quaternions context, Adv. Appl. Clifford Algebras 27, 2795-2803, 2017.
  • [46] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, The OEIS Foundation, https://oeis.org, 2021.
  • [47] A. Szynal-Liana and I. Włoch, The Pell quaternions and the Pell octonions, Adv. Appl. Clifford Algebras 26, 435-440, 2016.
  • [48] A. Szynal-Liana and I. Włoch, A note on Jacobsthal quaternions, Adv. Appl. Clifford Algebras 26, 441-447, 2016.
  • [49] E. Tan and H.-H. Leung, Some results on Horadam quaternions, Chaos Soliton Fract. 138, article 109961, 2020.
  • [50] D. Tasci, On k-Jacobsthal and k-Jacobsthal-Lucas quaternions, Journal of Science and Arts 3, 469-476, 2017.
  • [51] U. Tokeşer, Z. Ünal and G. Bilgici, Split Pell and Pell-Lucas quaternions, Adv. Appl. Clifford Algebras 27, 1881-1893, 2017.
  • [52] S. Vajda, Fibonacci & Lucas numbers, and the golden section, Ellis Horwood Ltd., Chichester, England, 1989.
  • [53] R. Vieira, F. Alves and P. Catarino, Relações bidimensionais e identidades da sequência de Leonardo, Revista Sergipana de Matemática e Educação Matemática 4, 156-173, 2019.
  • [54] R. Vieira, M. Mangueira, F. Alves and P. Catarino, A forma matricial dos números de Leonardo, Ciência e Natura 42, article e100, 2020.
  • [55] T. Yağmur, Split Jacobsthal and Jacobsthal-Lucas quaternions, Commun. Math. Appl. 10, 429-438, 2019.
There are 55 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Patrícia Beites 0000-0003-0266-7055

Paula Maria Machado Cruz Catarino 0000-0001-6917-5093

Project Number UIDB/00212/2020, MTM2017-83506-C2-2-P, UIDB/00013/2020, UIDP/00013/2020, UID/CED/00194/2020
Early Pub Date March 5, 2024
Publication Date August 27, 2024
Published in Issue Year 2024 Volume: 53 Issue: 4

Cite

APA Beites, P., & Catarino, P. M. M. C. (2024). On the Leonardo quaternions sequence. Hacettepe Journal of Mathematics and Statistics, 53(4), 1001-1023. https://doi.org/10.15672/hujms.1197693
AMA Beites P, Catarino PMMC. On the Leonardo quaternions sequence. Hacettepe Journal of Mathematics and Statistics. August 2024;53(4):1001-1023. doi:10.15672/hujms.1197693
Chicago Beites, Patrícia, and Paula Maria Machado Cruz Catarino. “On the Leonardo Quaternions Sequence”. Hacettepe Journal of Mathematics and Statistics 53, no. 4 (August 2024): 1001-23. https://doi.org/10.15672/hujms.1197693.
EndNote Beites P, Catarino PMMC (August 1, 2024) On the Leonardo quaternions sequence. Hacettepe Journal of Mathematics and Statistics 53 4 1001–1023.
IEEE P. Beites and P. M. M. C. Catarino, “On the Leonardo quaternions sequence”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 4, pp. 1001–1023, 2024, doi: 10.15672/hujms.1197693.
ISNAD Beites, Patrícia - Catarino, Paula Maria Machado Cruz. “On the Leonardo Quaternions Sequence”. Hacettepe Journal of Mathematics and Statistics 53/4 (August 2024), 1001-1023. https://doi.org/10.15672/hujms.1197693.
JAMA Beites P, Catarino PMMC. On the Leonardo quaternions sequence. Hacettepe Journal of Mathematics and Statistics. 2024;53:1001–1023.
MLA Beites, Patrícia and Paula Maria Machado Cruz Catarino. “On the Leonardo Quaternions Sequence”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 4, 2024, pp. 1001-23, doi:10.15672/hujms.1197693.
Vancouver Beites P, Catarino PMMC. On the Leonardo quaternions sequence. Hacettepe Journal of Mathematics and Statistics. 2024;53(4):1001-23.

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