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On the Leonardo Quaternions Sequence

Yıl 2024, Erken Görünüm, 1 - 23

Patrícia Beites Paula Maria Machado Cruz Catarino

Öz

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, Cata- lan’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-Ces`aro’s identity is also stated and proved. Finally, the generating function, the exponential generating function and the Poisson generating func- tion are deduced. In addition to the results on Leonardo quaternions, known results on Leonardo numbers and on Fibonacci quaternions are extended.

Anahtar Kelimeler

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

Destekleyen Kurum

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

Proje Numarası

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

Teşekkür

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.

Kaynakça

  • \bibitem{A} B. Aloui, A. Boussayoud, Generating functions of the product of the k-Fibonacci and k-Pell numbers and Chebyshev polynomials of the third and fourth kind, \textit{Mathematics in Engineering, Science and Aerospace} 12 (1) (2021).
  • \bibitem{AV} F. Alves, R. Vieira, The Newton fractal's Leonardo sequence study with the Google Colab, \textit{International Electronic Journal of Mathematics Education} 15 (2) (2020), article em0575.
  • \bibitem{Ak} M. Akyi\v{g}it, H. H. K\"{o}sal, M. Tosun, Split Fibonacci quaternions, \textit{Advances in Applied Clifford Algebras} {23} (2013), 535--545.
  • \bibitem{Ak1} M. Akyi\v{g}it, H. H. K\"{o}sal, M. Tosun, Fibonacci generalized quaternions, \textit{Advances in Applied Clifford Algebras} 24 (2014), 631--641.
  • \bibitem{sobreLeon} Y. Alp, E. G. Ko\c{c}er, Some properties of Leonardo numbers, \textit{Konuralp Journal of Mathematics} 9 (1) (2021), 183--189.
  • \bibitem{BNSV} P. D. Beites, A. P. Nicol\'{a}s, P. Saraiva, J. Vit\'{o}ria, Vector cross product differential and difference equations in $\mathbb{R}^3$ and in $\mathbb{R}^7$, \textit{Electronic Journal of Linear Algebra} 34 (2018), 675--686.
  • \bibitem{BN2016} P. D. Beites, A. P. Nicol\'{a}s, An associative triple system of the second kind, \textit{Communications in Algebra} 44 (11) (2016), 5027--5043.
  • \bibitem{BN2017} P. D. Beites, A. P. Nicol\'{a}s, A note on standard composition algebras of types II and III, \textit{Advances in Applied Clifford Algebras} 27 (2) (2017), 955--964.
  • \bibitem{Bi} G. Bilgici, U. Toke\c{s}er, Z. \"{U}nal, $k$-Fibonacci and $k$-Lucas generalized quaternions, \textit{Konuralp Journal of Mathematics} 5 (2017), 102--113.
  • \bibitem{BiC} G. Bilgici, P. Catarino, Unrestricted pell and pell-lucas quaternions, \textit{International Journal of Mathematics and Systems Science} 1 (3) (2018), article 816.
  • \bibitem{Terrence} T. R. Blackman, S. Lemurell, Spectral correspondences for Maass waveforms on quaternion groups, \textit{Journal of Number Theory} 158 (2016), 1--22.
  • \bibitem{CDNN2002} N. D. Cahill, J. R. D'Errico, D. A. Narayan, J. Y. Narayan, Fibonacci Determinants, \textit{The College Mathematics Journal} 33 (3) (2002), 221--225.
  • \bibitem{CMT} I. Ca\c{c}\~{a}o, H. R. Malonek, G. Tomaz, Shifted Generalized Pascal Matrices in the Context of Clifford Algebra-Valued Polynomial Sequences. In: Gervasi O. et al. (eds) Computational Science and Its Applications -- ICCSA 2017. Lecture Notes in Computer Science, Springer, 2017.
  • \bibitem{Ca} P. Catarino, The modified Pell and the modified $k$-Pell quaternions and octonions, \textit{Advances in Applied Clifford Algebras} 26 (2016), 577--590. \bibitem{CB1} P. Catarino, A. Borges, On Leonardo numbers, \textit{Acta Mathematica Universitatis Comenianae} 89 (1) (2020), 75--86.
  • \bibitem{CB2} P. Catarino, A. Borges, A note on incomplete Leonardo numbers, \textit{Integers} 20 (2020), article A43.
  • \bibitem{Hacettepe} P. Catarino, H. Campos, Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell numbers, \textit{Hacettepe Journal of Mathematics and Statistics}, 46 (3) (2017), 361-372. \bibitem{CatAlm} P. Catarino, R. De Almeida, On a quaternionic sequence with Vietoris' numbers, \textit{Filomat} 35 (4) (2021), in press.
  • \bibitem{CatAlmMediterranean} P. Catarino, R. De Almeida, A note on Vietoris' number sequence, \textit{Mediterranean Journal of Mathematics} 19 (1) (2022), 1--19.
  • \bibitem{yo} Y. Choo, A generalized quaternion with generalized Fibonacci number components, \textit{Applied Mathematical Sciences} 14 (1) (2020), 31--38.
  • \bibitem{Ci} C. B. \c{C}imen, A. \.{I}pek, On Pell quaternions and Pell-Lucas quaternions, \textit{Advances in Applied Clifford Algebras} 26 (2016), 39--51.
  • \bibitem{ConwayS} J. H. Conway, D. A. Smith, \textit{On Quaternions and Octonions: their Geometry, Arithmetic and Symmetry}, A. K. Peters, 2003.
  • \bibitem{CP} N. Correia, R. Pacheco, Harmonic maps of finite uniton number and their canonical elements, \textit{Annals of Global Analysis and Geometry} 47 (2015), 335--358.
  • \bibitem{Dasdemir} A. Da\c{s}demir, Gelin-Ces\`{a}ro identities for Fibonacci and Lucas quaternions, \textit{Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica} XVIII (2019), 137--144.
  • \bibitem{DSriv} G. B. Djordjevi\'{c}, H. M. Srivastava, Some generalizations of certain sequences associated with the Fibonacci numbers, \textit{Journal of the Indonesian Mathematical Society} 12 (2006), 99--112.
  • \bibitem{Dijkstra} E. W. Dijkstra, Smoothsort, an alternative for sorting in situ, \textit{Science of Computer Programming} 1 (1982), 223--233.
  • \bibitem{FMSS} M. I. Falc\~{a}o, F. Miranda, R. Severino, M. J. Soares, Evaluation schemes in the ring of quaternionic polynomials, \textit{BIT Numerical Mathematics} 58 (2018), 51--72.
  • \bibitem{Jacobson} N. Jacobson, Composition algebras and their automorphisms, \textit{Rendiconti del Circolo Matematico di Palermo} 7 (1958), 55--80.
  • \bibitem{Ha} S. Halici, On Fibonacci quaternions, \textit{Advances in Applied Clifford Algebras} 22 (2012), 321--327.
  • \bibitem{HaC} S. Halici, G. Cerda-Morales, On Quaternion-Gaussian Fibonacci numbers and their properties, \textit{Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica} 29 (1) (2021), 71--82.
  • \bibitem{HaKa} S. Halici, A. Karata\c{s}, On a generalization for Fibonacci quaternions, \textit{Chaos, Solitons and Fractals} {98} (2017), 178--182.
  • \bibitem{Horadam} A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, \textit{The American Mathematical Monthly} 70 (3) (1963), 289--291.
  • \bibitem{KTH2010} E. Kilic, D. Tasci, P. Haukkanen, On the generalized Lucas sequences by Hessenberg matrices, \textit{Ars Combinatoria} 95 (2010), 383--395.
  • \bibitem{Knuth} D. Knuth, \textit{The art of computer programming}, Addison Wesley Longman, 1997.
  • \bibitem{Koshy} T. Koshy, \textit{Fibonacci and Lucas numbers with applications}, Wiley, 2018.
  • \bibitem{Ipek} A. \.{I}pek, On $(p, q)$-Fibonacci quaternions and their Binet formulas, generating functions and certain binomial sums, \textit{Advances in Applied Clifford Algebras} 27 (2017), 1343--1351.
  • \bibitem{Leite} F. S. Leite, The geometry of hypercomplex matrices, \textit{Linear and Multilinear Algebra} 34 (2) (1993), 123--132.
  • \bibitem{MC} J. Morais, I. Ca\c c\~ao, Quaternion Zernike spherical polynomials, \textit{Mathematics of Computation} 84 (293) (2015), 1317--1337.
  • \bibitem{Pat} B. K. Patel, P. K. Ray, On the properties of $(p, q)$-Fibonacci and $(p, q)$-Lucas quaternions, \textit{Mathematical Reports} 21 (71) (2019), 15--25.
  • \bibitem{PolathKesim} E. Polath, S. Kesim, A note on Catalan's identity for the $k$-Fibonacci quaternions, \textit{Journal of Integer Sequences} 18 (2015), article 15.8.2.
  • \bibitem{P} E. Polatli, C. Kizilates, S. Kesim, On split $k$-Fibonacci and $k$-Lucas quaternions, \textit{Advances in Applied Clifford Algebras} 26 (2016), 353--362.
  • \bibitem{PSoykan} E. Polath, Y. Soykan, On Generalized Third-order Jacobsthal numbers, \textit{Asian Research Journal of Mathematics} 17 (2021), 1--19.
  • \bibitem{Ra} J. L. Ram\'{\i}rez, Some combinatorial properties of the $k$-Fibonacci and the $k$-Lucas quaternions, \textit{Analele Stiintifice ale Universitatii Ovidius Constanta} 23 (2) (2015), 201--212.
  • \bibitem{SBVK} N. Saba, A. Boussayoud, K. V. Kanuri, Mersenne Lucas numbers and complete homogeneous symmetric functions, \textit{Journal of Mathematics and Computer Science} 24 (2022), 127--139.
  • \bibitem{SBV} R. Ser\^{o}dio, P. D. Beites, J. Vit\'{o}ria, Intersection of a double cone and a line in the split-quaternions context, \textit{Advances in Applied Clifford algebras} 27 (2017), 2795--2803.
  • \bibitem{Sloane} N. J. A. Sloane, \textit{The On-Line Encyclopedia of Integer Sequences}, The OEIS Foundation, https://oeis.org (2021).
  • \bibitem{SzW} A. Szynal-Liana, I. W{\l}och, The Pell quaternions and the Pell octonions, \textit{Advances in Applied Clifford Algebras} 26 (2016), 435--440.
  • \bibitem{SzW1} A. Szynal-Liana, I. W{\l}och, A note on Jacobsthal quaternions, \textit{Advances in Applied Clifford Algebras} 26 (2016), 441--447.
  • \bibitem{TanLeung} E. Tan, H.-H. Leung, Some results on Horadam quaternions, \textit{Chaos, Solitons and Fractals} 138 (2020), article 109961.
  • \bibitem{T} D. Tasci, On $k$-Jacobsthal and $k$-Jacobsthal-Lucas quaternions, \textit{Journal of Science and Arts} 3 (40) (2017), 469--476.
  • \bibitem{Tok} U. Toke\c{s}er, Z. \"{U}nal, G. Bilgici, Split Pell and Pell-Lucas quaternions, \textit{Advances in Applied Clifford Algebras} 27 (2017), 1881--1893. \bibitem{Vajda} S. Vajda, \textit{Fibonacci \& Lucas numbers, and the golden section}, Ellis Horwood, 1989.
  • \bibitem{VACa} R. Vieira, F. Alves, P. Catarino, Rela\c{c}\~{o}es bidimensionais e identidades da sequ\^{e}ncia de Leonardo, \textit{Revista Sergipana de Matem\'{a}tica e Educa\c{c}\~{a}o Matem\'{a}tica} (2) (2019), 156--173.
  • \bibitem{VMACa} R. Vieira, M. Mangueira, F. Alves, P. Catarino, A forma matricial dos n\'{u}meros de Leonardo, \textit{Ci\^{e}ncia e Natura} 42 (2020), article e100.
  • \bibitem{y} T. Ya\v{g}mur, Split Jacobsthal and Jacobsthal-Lucas quaternions, \textit{Communications in Mathematics and Applications} 10 (3) (2019), 429--438.

Yıl 2024, Erken Görünüm, 1 - 23

Patrícia Beites Paula Maria Machado Cruz Catarino

Öz

Proje Numarası

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

Kaynakça

  • \bibitem{A} B. Aloui, A. Boussayoud, Generating functions of the product of the k-Fibonacci and k-Pell numbers and Chebyshev polynomials of the third and fourth kind, \textit{Mathematics in Engineering, Science and Aerospace} 12 (1) (2021).
  • \bibitem{AV} F. Alves, R. Vieira, The Newton fractal's Leonardo sequence study with the Google Colab, \textit{International Electronic Journal of Mathematics Education} 15 (2) (2020), article em0575.
  • \bibitem{Ak} M. Akyi\v{g}it, H. H. K\"{o}sal, M. Tosun, Split Fibonacci quaternions, \textit{Advances in Applied Clifford Algebras} {23} (2013), 535--545.
  • \bibitem{Ak1} M. Akyi\v{g}it, H. H. K\"{o}sal, M. Tosun, Fibonacci generalized quaternions, \textit{Advances in Applied Clifford Algebras} 24 (2014), 631--641.
  • \bibitem{sobreLeon} Y. Alp, E. G. Ko\c{c}er, Some properties of Leonardo numbers, \textit{Konuralp Journal of Mathematics} 9 (1) (2021), 183--189.
  • \bibitem{BNSV} P. D. Beites, A. P. Nicol\'{a}s, P. Saraiva, J. Vit\'{o}ria, Vector cross product differential and difference equations in $\mathbb{R}^3$ and in $\mathbb{R}^7$, \textit{Electronic Journal of Linear Algebra} 34 (2018), 675--686.
  • \bibitem{BN2016} P. D. Beites, A. P. Nicol\'{a}s, An associative triple system of the second kind, \textit{Communications in Algebra} 44 (11) (2016), 5027--5043.
  • \bibitem{BN2017} P. D. Beites, A. P. Nicol\'{a}s, A note on standard composition algebras of types II and III, \textit{Advances in Applied Clifford Algebras} 27 (2) (2017), 955--964.
  • \bibitem{Bi} G. Bilgici, U. Toke\c{s}er, Z. \"{U}nal, $k$-Fibonacci and $k$-Lucas generalized quaternions, \textit{Konuralp Journal of Mathematics} 5 (2017), 102--113.
  • \bibitem{BiC} G. Bilgici, P. Catarino, Unrestricted pell and pell-lucas quaternions, \textit{International Journal of Mathematics and Systems Science} 1 (3) (2018), article 816.
  • \bibitem{Terrence} T. R. Blackman, S. Lemurell, Spectral correspondences for Maass waveforms on quaternion groups, \textit{Journal of Number Theory} 158 (2016), 1--22.
  • \bibitem{CDNN2002} N. D. Cahill, J. R. D'Errico, D. A. Narayan, J. Y. Narayan, Fibonacci Determinants, \textit{The College Mathematics Journal} 33 (3) (2002), 221--225.
  • \bibitem{CMT} I. Ca\c{c}\~{a}o, H. R. Malonek, G. Tomaz, Shifted Generalized Pascal Matrices in the Context of Clifford Algebra-Valued Polynomial Sequences. In: Gervasi O. et al. (eds) Computational Science and Its Applications -- ICCSA 2017. Lecture Notes in Computer Science, Springer, 2017.
  • \bibitem{Ca} P. Catarino, The modified Pell and the modified $k$-Pell quaternions and octonions, \textit{Advances in Applied Clifford Algebras} 26 (2016), 577--590. \bibitem{CB1} P. Catarino, A. Borges, On Leonardo numbers, \textit{Acta Mathematica Universitatis Comenianae} 89 (1) (2020), 75--86.
  • \bibitem{CB2} P. Catarino, A. Borges, A note on incomplete Leonardo numbers, \textit{Integers} 20 (2020), article A43.
  • \bibitem{Hacettepe} P. Catarino, H. Campos, Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell numbers, \textit{Hacettepe Journal of Mathematics and Statistics}, 46 (3) (2017), 361-372. \bibitem{CatAlm} P. Catarino, R. De Almeida, On a quaternionic sequence with Vietoris' numbers, \textit{Filomat} 35 (4) (2021), in press.
  • \bibitem{CatAlmMediterranean} P. Catarino, R. De Almeida, A note on Vietoris' number sequence, \textit{Mediterranean Journal of Mathematics} 19 (1) (2022), 1--19.
  • \bibitem{yo} Y. Choo, A generalized quaternion with generalized Fibonacci number components, \textit{Applied Mathematical Sciences} 14 (1) (2020), 31--38.
  • \bibitem{Ci} C. B. \c{C}imen, A. \.{I}pek, On Pell quaternions and Pell-Lucas quaternions, \textit{Advances in Applied Clifford Algebras} 26 (2016), 39--51.
  • \bibitem{ConwayS} J. H. Conway, D. A. Smith, \textit{On Quaternions and Octonions: their Geometry, Arithmetic and Symmetry}, A. K. Peters, 2003.
  • \bibitem{CP} N. Correia, R. Pacheco, Harmonic maps of finite uniton number and their canonical elements, \textit{Annals of Global Analysis and Geometry} 47 (2015), 335--358.
  • \bibitem{Dasdemir} A. Da\c{s}demir, Gelin-Ces\`{a}ro identities for Fibonacci and Lucas quaternions, \textit{Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica} XVIII (2019), 137--144.
  • \bibitem{DSriv} G. B. Djordjevi\'{c}, H. M. Srivastava, Some generalizations of certain sequences associated with the Fibonacci numbers, \textit{Journal of the Indonesian Mathematical Society} 12 (2006), 99--112.
  • \bibitem{Dijkstra} E. W. Dijkstra, Smoothsort, an alternative for sorting in situ, \textit{Science of Computer Programming} 1 (1982), 223--233.
  • \bibitem{FMSS} M. I. Falc\~{a}o, F. Miranda, R. Severino, M. J. Soares, Evaluation schemes in the ring of quaternionic polynomials, \textit{BIT Numerical Mathematics} 58 (2018), 51--72.
  • \bibitem{Jacobson} N. Jacobson, Composition algebras and their automorphisms, \textit{Rendiconti del Circolo Matematico di Palermo} 7 (1958), 55--80.
  • \bibitem{Ha} S. Halici, On Fibonacci quaternions, \textit{Advances in Applied Clifford Algebras} 22 (2012), 321--327.
  • \bibitem{HaC} S. Halici, G. Cerda-Morales, On Quaternion-Gaussian Fibonacci numbers and their properties, \textit{Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica} 29 (1) (2021), 71--82.
  • \bibitem{HaKa} S. Halici, A. Karata\c{s}, On a generalization for Fibonacci quaternions, \textit{Chaos, Solitons and Fractals} {98} (2017), 178--182.
  • \bibitem{Horadam} A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, \textit{The American Mathematical Monthly} 70 (3) (1963), 289--291.
  • \bibitem{KTH2010} E. Kilic, D. Tasci, P. Haukkanen, On the generalized Lucas sequences by Hessenberg matrices, \textit{Ars Combinatoria} 95 (2010), 383--395.
  • \bibitem{Knuth} D. Knuth, \textit{The art of computer programming}, Addison Wesley Longman, 1997.
  • \bibitem{Koshy} T. Koshy, \textit{Fibonacci and Lucas numbers with applications}, Wiley, 2018.
  • \bibitem{Ipek} A. \.{I}pek, On $(p, q)$-Fibonacci quaternions and their Binet formulas, generating functions and certain binomial sums, \textit{Advances in Applied Clifford Algebras} 27 (2017), 1343--1351.
  • \bibitem{Leite} F. S. Leite, The geometry of hypercomplex matrices, \textit{Linear and Multilinear Algebra} 34 (2) (1993), 123--132.
  • \bibitem{MC} J. Morais, I. Ca\c c\~ao, Quaternion Zernike spherical polynomials, \textit{Mathematics of Computation} 84 (293) (2015), 1317--1337.
  • \bibitem{Pat} B. K. Patel, P. K. Ray, On the properties of $(p, q)$-Fibonacci and $(p, q)$-Lucas quaternions, \textit{Mathematical Reports} 21 (71) (2019), 15--25.
  • \bibitem{PolathKesim} E. Polath, S. Kesim, A note on Catalan's identity for the $k$-Fibonacci quaternions, \textit{Journal of Integer Sequences} 18 (2015), article 15.8.2.
  • \bibitem{P} E. Polatli, C. Kizilates, S. Kesim, On split $k$-Fibonacci and $k$-Lucas quaternions, \textit{Advances in Applied Clifford Algebras} 26 (2016), 353--362.
  • \bibitem{PSoykan} E. Polath, Y. Soykan, On Generalized Third-order Jacobsthal numbers, \textit{Asian Research Journal of Mathematics} 17 (2021), 1--19.
  • \bibitem{Ra} J. L. Ram\'{\i}rez, Some combinatorial properties of the $k$-Fibonacci and the $k$-Lucas quaternions, \textit{Analele Stiintifice ale Universitatii Ovidius Constanta} 23 (2) (2015), 201--212.
  • \bibitem{SBVK} N. Saba, A. Boussayoud, K. V. Kanuri, Mersenne Lucas numbers and complete homogeneous symmetric functions, \textit{Journal of Mathematics and Computer Science} 24 (2022), 127--139.
  • \bibitem{SBV} R. Ser\^{o}dio, P. D. Beites, J. Vit\'{o}ria, Intersection of a double cone and a line in the split-quaternions context, \textit{Advances in Applied Clifford algebras} 27 (2017), 2795--2803.
  • \bibitem{Sloane} N. J. A. Sloane, \textit{The On-Line Encyclopedia of Integer Sequences}, The OEIS Foundation, https://oeis.org (2021).
  • \bibitem{SzW} A. Szynal-Liana, I. W{\l}och, The Pell quaternions and the Pell octonions, \textit{Advances in Applied Clifford Algebras} 26 (2016), 435--440.
  • \bibitem{SzW1} A. Szynal-Liana, I. W{\l}och, A note on Jacobsthal quaternions, \textit{Advances in Applied Clifford Algebras} 26 (2016), 441--447.
  • \bibitem{TanLeung} E. Tan, H.-H. Leung, Some results on Horadam quaternions, \textit{Chaos, Solitons and Fractals} 138 (2020), article 109961.
  • \bibitem{T} D. Tasci, On $k$-Jacobsthal and $k$-Jacobsthal-Lucas quaternions, \textit{Journal of Science and Arts} 3 (40) (2017), 469--476.
  • \bibitem{Tok} U. Toke\c{s}er, Z. \"{U}nal, G. Bilgici, Split Pell and Pell-Lucas quaternions, \textit{Advances in Applied Clifford Algebras} 27 (2017), 1881--1893. \bibitem{Vajda} S. Vajda, \textit{Fibonacci \& Lucas numbers, and the golden section}, Ellis Horwood, 1989.
  • \bibitem{VACa} R. Vieira, F. Alves, P. Catarino, Rela\c{c}\~{o}es bidimensionais e identidades da sequ\^{e}ncia de Leonardo, \textit{Revista Sergipana de Matem\'{a}tica e Educa\c{c}\~{a}o Matem\'{a}tica} (2) (2019), 156--173.
  • \bibitem{VMACa} R. Vieira, M. Mangueira, F. Alves, P. Catarino, A forma matricial dos n\'{u}meros de Leonardo, \textit{Ci\^{e}ncia e Natura} 42 (2020), article e100.
  • \bibitem{y} T. Ya\v{g}mur, Split Jacobsthal and Jacobsthal-Lucas quaternions, \textit{Communications in Mathematics and Applications} 10 (3) (2019), 429--438.
Toplam 52 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Patrícia Beites 0000-0003-0266-7055

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

Proje Numarası UIDB/00212/2020, MTM2017-83506-C2-2-P, UIDB/00013/2020, UIDP/00013/2020, UID/CED/00194/2020
Erken Görünüm Tarihi 5 Mart 2024
Yayımlanma Tarihi
Yayımlandığı Sayı Yıl 2024 Erken Görünüm

Kaynak Göster

APA Beites, P., & Catarino, P. M. M. C. (2024). On the Leonardo Quaternions Sequence. Hacettepe Journal of Mathematics and Statistics1-23.
AMA Beites P, Catarino PMMC. On the Leonardo Quaternions Sequence. Hacettepe Journal of Mathematics and Statistics. Published online 01 Mart 2024:1-23.
Chicago Beites, Patrícia, ve Paula Maria Machado Cruz Catarino. “On the Leonardo Quaternions Sequence”. Hacettepe Journal of Mathematics and Statistics, Mart (Mart 2024), 1-23.
EndNote Beites P, Catarino PMMC (01 Mart 2024) On the Leonardo Quaternions Sequence. Hacettepe Journal of Mathematics and Statistics 1–23.
IEEE P. Beites ve P. M. M. C. Catarino, “On the Leonardo Quaternions Sequence”, Hacettepe Journal of Mathematics and Statistics, ss. 1–23, Mart 2024.
ISNAD Beites, Patrícia - Catarino, Paula Maria Machado Cruz. “On the Leonardo Quaternions Sequence”. Hacettepe Journal of Mathematics and Statistics. Mart 2024. 1-23.
JAMA Beites P, Catarino PMMC. On the Leonardo Quaternions Sequence. Hacettepe Journal of Mathematics and Statistics. 2024;:1–23.
MLA Beites, Patrícia ve Paula Maria Machado Cruz Catarino. “On the Leonardo Quaternions Sequence”. Hacettepe Journal of Mathematics and Statistics, 2024, ss. 1-23.
Vancouver Beites P, Catarino PMMC. On the Leonardo Quaternions Sequence. Hacettepe Journal of Mathematics and Statistics. 2024:1-23.

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