Geodesic
On Balancing Quaternions and Lucas-Balancing Quaternions
Discussiones Mathematicae. General Algebra and Applications, Tome 41 (2021) no. 1, pp. 55-68.

Voir la notice de l'article provenant de la source Library of Science

In this paper we define and study balancing quaternions and Lucas-balancing quaternions. We give the generating functions, matrix generators and Binet formulas for these numbers. Moreover, the well-known properties e.g. Catalan, d’ Ocagne identities have been obtained for these quaternions.
Keywords: balancing number, Lucas-balancing number, quaternion, Binet formula, generating function 
@article{DMGAA_2021_41_1_a5,
     author = {Br\'od, Dorota},
     title = {On {Balancing} {Quaternions} and {Lucas-Balancing} {Quaternions}},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {55--68},
     publisher = {mathdoc},
     volume = {41},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2021_41_1_a5/}
}
TY  - JOUR
AU  - Bród, Dorota
TI  - On Balancing Quaternions and Lucas-Balancing Quaternions
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2021
SP  - 55
EP  - 68
VL  - 41
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2021_41_1_a5/
LA  - en
ID  - DMGAA_2021_41_1_a5
ER  - 
%0 Journal Article
%A Bród, Dorota
%T On Balancing Quaternions and Lucas-Balancing Quaternions
%J Discussiones Mathematicae. General Algebra and Applications
%D 2021
%P 55-68
%V 41
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGAA_2021_41_1_a5/
%G en
%F DMGAA_2021_41_1_a5
Bród, Dorota. On Balancing Quaternions and Lucas-Balancing Quaternions. Discussiones Mathematicae. General Algebra and Applications, Tome 41 (2021) no. 1, pp. 55-68. http://geodesic.mathdoc.fr/item/DMGAA_2021_41_1_a5/

[1] A. Behera and G.K. Panda, On the square roots of triangular numbers, Fibonacci Quart. 37 (1999) 98–105.

[2] P. Catarino, H. Campos and P. Vasco, On some identities for balancing and cobalancing numbers, Ann. Math. Inform. 45 (2015) 11–24. http://ami.ektf.hu

[3] C.B. Çimen, A. İpek, On Pell Quaternions and Pell-Lucas Quaternions, Adv. Appl. Clifford Alg. 26 (2016) 39–51. doi:10.1007/s00006-015-0571-8

[4] A.F. Horadam, Complex Fibonacci Numbers and Fibonacci Quaternions, Amer. Math. Monthly 70 (1963) 289–291. doi:10.2307/2313129

[5] A. İpek, On ( p, q ) -Fibonacci quaternions and their Binet formulas, generating functions and certain binomial sums, Adv. Appl. Clifford Alg. 27 (2017) 1343–1351. doi:10.1007/s00006-016-0704-8

[6] G.K. Panda, Some fascinating properties of balancing numbers, Proc. Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium 194 (2009) 185–189.

[7] G.K. Panda and P.K. Ray, Cobalancing numbers and cobalancers, Int. J. Math. and Math. Sci. 8 (2005) 1189–1200. doi:10.1155/IJMMS.2005.1189

[8] B.K. Patel and P.K. Ray, On the properties of (p, q)-Fibonacci and (p, q)-Lucas quaternions, Math. Reports 21 (2019) 15–25.

[9] P.K. Ray, Certain Matrices Associated with Balancing and Lucas-balancing Numbers, Matematika 28 (2012) 15–22.

[10] P.K. Ray and J. Sahu, Generating functions for certain balancing and Lucas-balancing numbers, Palest. J. Math. 5 (2016) 122–129.

[11] A. Szynal-Liana and I. W loch, A note on Jacobsthal quaternions, Adv. Appl. Clifford Alg. 26 (2016) 441–447. doi:10.1007/s00006-015-0622-1

[12] A. Szynal-Liana and I. W loch, The Pell Quaternions and the Pell Octonions, Adv. Appl. Clifford Alg. 26 (2016) 435–440. doi:10.1007/s00006-015-0570-9

[13] J.P. Ward, Quaternions and Cayley Numbers: Algebra and Applications (Kluwer Academic Publishers, London, 1997).