International Journal of Nonlinear Analysis and Applications
A study on hyperbolic numbers with generalized Jacobsthal numbers components
Document Type : Research Paper
Authors
1 Department of Mathematics, Art and Science Faculty, Zonguldak Bulent Ecevit University, 67100, Zonguldak, Turkey
2 Kirklareli University, Pinarhisar Vocational School, 39300, Kirklareli, Turkey
Abstract
In this paper, we introduce the generalized hyperbolic Jacobsthal numbers. As special cases, we deal with hyperbolic Jacobsthal and hyperbolic Jacobsthal-Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences.
Keywords
[1] M. Akar, S. Y¨uce and S¸. S¸ahin, On the dual hyperbolic numbers and the complex hyperbolic numbers, J. Comput.
Sci. Comput.Math. 8 (2018), no. 1, 1–6.
[2] M. Akbulak and A. Otele¸s, ¨ On the sum of Pell and Jacobsthal numbers by matrix method, Bull. Iranian Math.
Soc. 40 (2014), no. 4, 1017–1025.
[3] F.T. Aydın, On generalizations of the Jacobsthal sequence, Notes Number Theory Discrete Math. 24 (2018), no.
1, 120–135.
[4] F.T. Aydın, Hyperbolic Fibonacci sequence, Universal J. Math. Appl. 2 (2019), no. 2, 59–64.
[5] J. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002), no. 2, 145–205.
[6] D.K. Biss, D. Dugger and D.C. Isaksen, Large annihilators in Cayley-Dickson algebras, Commun. Algebra 36
(2008), no. 2, 632–664.
[7] P. Catarino, P. Vasco, A.P.A. Campos and A. Borges, New families of Jacobsthal and Jacobsthal-Lucas numbers,
Algebra Discrete Math. 20 (2015) , no. 1, 40–54.
[8] Z. Cerin, ˇ Formulae for sums of Jacobsthal–Lucas numbers, Int. Math. Forum 2 (2007), no. 40, 1969–1984.
[9] Z. Cerin, ˇ Sums of Squares and Products of Jacobsthal Numbers, J. Integer Seq. 10 (2007), 1–15.[10] H.H. Cheng and S. Thompson, Dual Polynomials and complex dual numbers for analysis of spatial mechanisms,
Proc. ASME 24th Biennial Mechanisms Conference, Irvine, CA, August, 1996, pp. 19–22.
[11] J. Cockle, On a new imaginary in algebra, Phil. Mag. London-Dublin-Edin. 3 (1849), no. 34, 37–47.
[12] A. Dasdemir, On the Jacobsthal numbers by matrix method, SDU J. Sci. 7 (2012), no. 1, 69–76.
[13] A. Da¸sdemir, A study on the Jacobsthal and Jacobsthal–Lucas numbers by matrix method, DUFED J. Sci. 3
(2014), no. 1, 13–18.
[14] CM. Dikmen, Hyperbolic Jacobsthal Numbers, Asian Research Journal of Mathematics 15(4) (2019) 1-9.
[15] PG. Fjelstad and SG. Gal, n-dimensional Hyperbolic Complex Numbers, Adv. Appl. Clifford Algebras 8 (1998),
no. 1, 47–68.
[16] A. Gnanam and B. Anitha, Sums of Squares Jacobsthal Numbers, IOSR J. Math. 11 (2015), no. 6, 62–64.
[17] W.R. Hamilton, Elements of quaternions, Chelsea Publishing Company, New York, 1969.
[18] A.F. Horadam, Jacobsthal representation numbers, Fibonacci Quart. 34 (1996), 40–54.
[19] A.F. Horadam, Jacobsthal and Pell curves, Fibonacci Quart. 26 (1988), 77–83.
[20] K. Imaeda and M. Imaeda, Sedenions: Algebra and analysis, Appl. Math. Comput. 115 (2000), 77–88.
[21] B. Jancewicz, The extended Grassmann algebra of R3, Clifford (Geometric) Algebras with Applications and
Engineering, Birkhauser, Boston, 1996, 389-421.
[22] I. Kantor and A. Solodovnikov, Hypercomplex Numbers, Springer-Verlag, New York, 1989.
[23] A. Khrennikov and G. Segre, An Introduction to hyperbolic analysis, http://arxiv.org/abs/math-ph/0507053v2,
2005.
[24] G.E. Kocer, Circulant, negacyclic and semicirculant matrices with the modified Pell, Jacobsthal and JacobsthalLucas numbers, Hacet. J. Math. Stat. 36 (2007), no. 2, 133–142.
[25] F. K¨oken and D. Bozkurt, On the Jacobsthal numbers by matrix methods, Int. J. Contemp Math. Sciences 3(13)
(2008) 605-614.
[26] V.V. Kravchenko, Hyperbolic numbers and analytic functions, Applied Pseudoanalytic Function Theory, Frontiers
in Mathematics. Birkh¨auser Basel, 2009.
[27] V. Mazorchuk, New families of Jacobsthal and Jacobsthal-Lucas numbers, Algebra Discrete Math. 20 (2015), no.
1, 40–54.
[28] G. Moreno, The zero divisors of the Cayley-Dickson algebras over the real numbers, Bol. Soc. Mat. Mexicana 4
(1998), no. 3, 13–28.
[29] A.E. Motter and A.F. Rosa, Hyperbolic calculus, Adv. Appl. Clifford Algebr. 8 (1998), no. 1, 109–128.
[30] N.J.A. Sloane, The on-line encyclopedia of integer sequences, Available: http://oeis.org/
[31] G. Sobczyk, The hyperbolic number plane, College Math. J. 26 (1995), no. 4, 268–280.
[32] G. Sobczyk, Complex and hyperbolic numbers, New Foundations in Mathematics, Birkh¨auser, Boston, 2013.
[33] Y. Soykan, Tribonacci and Tribonacci-Lucas Sedenions, Math. 7 (2019), no. 1, 1–19.
[34] Y. Soykan, On summing formulas for generalized Fibonacci and Gaussian generalized Fibonacci numbers, Adv.
Res. 20 (2019), no. 2, 1–15.
[35] Y. Soykan, Corrigendum: On summing formulas for generalized Fibonacci and Gaussian generalized Fibonacci
numbers, 2019.
[36] Y. Soykan, On hyperbolic numbers with generalized Fibonacci numbers components, Researchgate Preprint, DOI:
10.13140/RG.2.2.19903.87207.
[37] S¸. Uygun, Some sum formulas of (s, t)-Jacobsthal and (s, t)-Jacobsthal Lucas matrix sequences, Appl. Math. 7
(2019), 61–69.
Sci. Comput.Math. 8 (2018), no. 1, 1–6.
[2] M. Akbulak and A. Otele¸s, ¨ On the sum of Pell and Jacobsthal numbers by matrix method, Bull. Iranian Math.
Soc. 40 (2014), no. 4, 1017–1025.
[3] F.T. Aydın, On generalizations of the Jacobsthal sequence, Notes Number Theory Discrete Math. 24 (2018), no.
1, 120–135.
[4] F.T. Aydın, Hyperbolic Fibonacci sequence, Universal J. Math. Appl. 2 (2019), no. 2, 59–64.
[5] J. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002), no. 2, 145–205.
[6] D.K. Biss, D. Dugger and D.C. Isaksen, Large annihilators in Cayley-Dickson algebras, Commun. Algebra 36
(2008), no. 2, 632–664.
[7] P. Catarino, P. Vasco, A.P.A. Campos and A. Borges, New families of Jacobsthal and Jacobsthal-Lucas numbers,
Algebra Discrete Math. 20 (2015) , no. 1, 40–54.
[8] Z. Cerin, ˇ Formulae for sums of Jacobsthal–Lucas numbers, Int. Math. Forum 2 (2007), no. 40, 1969–1984.
[9] Z. Cerin, ˇ Sums of Squares and Products of Jacobsthal Numbers, J. Integer Seq. 10 (2007), 1–15.[10] H.H. Cheng and S. Thompson, Dual Polynomials and complex dual numbers for analysis of spatial mechanisms,
Proc. ASME 24th Biennial Mechanisms Conference, Irvine, CA, August, 1996, pp. 19–22.
[11] J. Cockle, On a new imaginary in algebra, Phil. Mag. London-Dublin-Edin. 3 (1849), no. 34, 37–47.
[12] A. Dasdemir, On the Jacobsthal numbers by matrix method, SDU J. Sci. 7 (2012), no. 1, 69–76.
[13] A. Da¸sdemir, A study on the Jacobsthal and Jacobsthal–Lucas numbers by matrix method, DUFED J. Sci. 3
(2014), no. 1, 13–18.
[14] CM. Dikmen, Hyperbolic Jacobsthal Numbers, Asian Research Journal of Mathematics 15(4) (2019) 1-9.
[15] PG. Fjelstad and SG. Gal, n-dimensional Hyperbolic Complex Numbers, Adv. Appl. Clifford Algebras 8 (1998),
no. 1, 47–68.
[16] A. Gnanam and B. Anitha, Sums of Squares Jacobsthal Numbers, IOSR J. Math. 11 (2015), no. 6, 62–64.
[17] W.R. Hamilton, Elements of quaternions, Chelsea Publishing Company, New York, 1969.
[18] A.F. Horadam, Jacobsthal representation numbers, Fibonacci Quart. 34 (1996), 40–54.
[19] A.F. Horadam, Jacobsthal and Pell curves, Fibonacci Quart. 26 (1988), 77–83.
[20] K. Imaeda and M. Imaeda, Sedenions: Algebra and analysis, Appl. Math. Comput. 115 (2000), 77–88.
[21] B. Jancewicz, The extended Grassmann algebra of R3, Clifford (Geometric) Algebras with Applications and
Engineering, Birkhauser, Boston, 1996, 389-421.
[22] I. Kantor and A. Solodovnikov, Hypercomplex Numbers, Springer-Verlag, New York, 1989.
[23] A. Khrennikov and G. Segre, An Introduction to hyperbolic analysis, http://arxiv.org/abs/math-ph/0507053v2,
2005.
[24] G.E. Kocer, Circulant, negacyclic and semicirculant matrices with the modified Pell, Jacobsthal and JacobsthalLucas numbers, Hacet. J. Math. Stat. 36 (2007), no. 2, 133–142.
[25] F. K¨oken and D. Bozkurt, On the Jacobsthal numbers by matrix methods, Int. J. Contemp Math. Sciences 3(13)
(2008) 605-614.
[26] V.V. Kravchenko, Hyperbolic numbers and analytic functions, Applied Pseudoanalytic Function Theory, Frontiers
in Mathematics. Birkh¨auser Basel, 2009.
[27] V. Mazorchuk, New families of Jacobsthal and Jacobsthal-Lucas numbers, Algebra Discrete Math. 20 (2015), no.
1, 40–54.
[28] G. Moreno, The zero divisors of the Cayley-Dickson algebras over the real numbers, Bol. Soc. Mat. Mexicana 4
(1998), no. 3, 13–28.
[29] A.E. Motter and A.F. Rosa, Hyperbolic calculus, Adv. Appl. Clifford Algebr. 8 (1998), no. 1, 109–128.
[30] N.J.A. Sloane, The on-line encyclopedia of integer sequences, Available: http://oeis.org/
[31] G. Sobczyk, The hyperbolic number plane, College Math. J. 26 (1995), no. 4, 268–280.
[32] G. Sobczyk, Complex and hyperbolic numbers, New Foundations in Mathematics, Birkh¨auser, Boston, 2013.
[33] Y. Soykan, Tribonacci and Tribonacci-Lucas Sedenions, Math. 7 (2019), no. 1, 1–19.
[34] Y. Soykan, On summing formulas for generalized Fibonacci and Gaussian generalized Fibonacci numbers, Adv.
Res. 20 (2019), no. 2, 1–15.
[35] Y. Soykan, Corrigendum: On summing formulas for generalized Fibonacci and Gaussian generalized Fibonacci
numbers, 2019.
[36] Y. Soykan, On hyperbolic numbers with generalized Fibonacci numbers components, Researchgate Preprint, DOI:
10.13140/RG.2.2.19903.87207.
[37] S¸. Uygun, Some sum formulas of (s, t)-Jacobsthal and (s, t)-Jacobsthal Lucas matrix sequences, Appl. Math. 7
(2019), 61–69.
)
Volume 13, Issue 2
July 2022Pages 1965-1981
- Receive Date: 15 December 2020
- Revise Date: 30 May 2021
- Accept Date: 12 June 2021