Generalization of Cassini formulas for balancing and Lucas-balancing
numbers

P. K. Ray; K. Parida

The mathematical identity that connects three adjacent balancing numbers is well known under the name Cassini formula, and is used to establish many important identities involving balancing numbers and their related sequences. This article is an attempt to draw attention to some of the unusual properties of generalized balancing numbers, in particular, to the general- ized Cassini formula.

1. A. Behera, G.K. Panda, On the square roots of triangular numbers, Fibonacci Quarterly, 37 (1999), №2, 98–105.

2. A. Berczes, K. Liptai, I. Pink, On generalized balancing numbers, Fibonacci Quarterly, 48 (2010), №2, 121–128.

3. R. Keskin, O. Karaatly, Some new properties of balancing numbers and square triangular numbers, Journal of Integer Sequences, 15 (2012), №1.

4. K. Liptai, F. Luca, A. Pinter, L. Szalay, Generalized balancing numbers, Indagationes Mathematicae, 20 (2009), 87–100.

5. P. Olajos, Properties of balancing, cobalancing and generalized balancing numbers, Annales Mathematicae et Informaticae, 37 (2010), 125–138.

6. G.K. Panda, P.K. Ray, Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 6 (2011), №1, 41–72.

7. G.K. Panda, P.K. Ray, Cobalancing numbers and cobalancers, International Journal of Mathematics and Mathematical Sciences, 8 (2005), 1189–1200.

8. G.K. Panda, Some fascinating properties of balancing numbers, Proceeding of the eleventh international conference on Fibonacci numbers and their applications, Cong. Numerantium, 194 (2009), 185–189.

9. G.K. Panda, S.S. Rout, Gap balancing numbers, Fibonacci Quarterly, 51 (2013), №3, 239–248.

10. P.K. Ray, Application of Chybeshev polynomials in factorization of balancing and Lucas-balancing numbers, Boletim Da Sociedade Paranaense De Matematica, 30 (2012), №2, 49–56.

11. P.K. Ray, Factorization of negatively subscripted balancing and Lucas-balancing numbers, Boletim Da Sociedade Paranaense De Matematica, 31 (2013), №2, 161–173.

12. P.K. Ray, Curious congruences for balancing numbers, International Journal of Contemporary Mathematical Sciences, 7 (2012), №18, 881–889.

13. P.K. Ray, New identities for the common factors of balancing and Lucas-balancing numbers, International Journal of Pure and Applied Mathematics, 85 (2013), 487–494.

14. P.K. Ray, Some congruences for balancing and Lucas-balancing numbers and their applications, Integers, 14 (2014), #A8.

15. P.K. Ray, On the properties of Lucas-balancing numbers by matrix method, Sigmae, Alfenas, 3 (2014), №1, 1–6.

16. P.K. Ray, Balancing sequences of matrices with application to algebra of balancing numbers, Notes on number theory and discrete mathematics, 20 (2014), №1, 49–58.

	Generalization of Cassini formulas for balancing and Lucas-balancing numbers
Author	P. K. Ray, K. Parida prasanta@iiit-bh.ac.in, parida.kaberi@gmail.com National Institute of Technology, Rourkela, India
Abstract	The mathematical identity that connects three adjacent balancing numbers is well known under the name Cassini formula, and is used to establish many important identities involving balancing numbers and their related sequences. This article is an attempt to draw attention to some of the unusual properties of generalized balancing numbers, in particular, to the general- ized Cassini formula.
Keywords	balancing numbers; Lucas-balancing numbers; Cassini formula
Reference	1. A. Behera, G.K. Panda, On the square roots of triangular numbers, Fibonacci Quarterly, 37 (1999), №2, 98–105. 2. A. Berczes, K. Liptai, I. Pink, On generalized balancing numbers, Fibonacci Quarterly, 48 (2010), №2, 121–128. 3. R. Keskin, O. Karaatly, Some new properties of balancing numbers and square triangular numbers, Journal of Integer Sequences, 15 (2012), №1. 4. K. Liptai, F. Luca, A. Pinter, L. Szalay, Generalized balancing numbers, Indagationes Mathematicae, 20 (2009), 87–100. 5. P. Olajos, Properties of balancing, cobalancing and generalized balancing numbers, Annales Mathematicae et Informaticae, 37 (2010), 125–138. 6. G.K. Panda, P.K. Ray, Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 6 (2011), №1, 41–72. 7. G.K. Panda, P.K. Ray, Cobalancing numbers and cobalancers, International Journal of Mathematics and Mathematical Sciences, 8 (2005), 1189–1200. 8. G.K. Panda, Some fascinating properties of balancing numbers, Proceeding of the eleventh international conference on Fibonacci numbers and their applications, Cong. Numerantium, 194 (2009), 185–189. 9. G.K. Panda, S.S. Rout, Gap balancing numbers, Fibonacci Quarterly, 51 (2013), №3, 239–248. 10. P.K. Ray, Application of Chybeshev polynomials in factorization of balancing and Lucas-balancing numbers, Boletim Da Sociedade Paranaense De Matematica, 30 (2012), №2, 49–56. 11. P.K. Ray, Factorization of negatively subscripted balancing and Lucas-balancing numbers, Boletim Da Sociedade Paranaense De Matematica, 31 (2013), №2, 161–173. 12. P.K. Ray, Curious congruences for balancing numbers, International Journal of Contemporary Mathematical Sciences, 7 (2012), №18, 881–889. 13. P.K. Ray, New identities for the common factors of balancing and Lucas-balancing numbers, International Journal of Pure and Applied Mathematics, 85 (2013), 487–494. 14. P.K. Ray, Some congruences for balancing and Lucas-balancing numbers and their applications, Integers, 14 (2014), #A8. 15. P.K. Ray, On the properties of Lucas-balancing numbers by matrix method, Sigmae, Alfenas, 3 (2014), №1, 1–6. 16. P.K. Ray, Balancing sequences of matrices with application to algebra of balancing numbers, Notes on number theory and discrete mathematics, 20 (2014), №1, 49–58.
Pages	9-14
Volume	42
Issue	1
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Generalization of Cassini formulas for balancing and Lucas-balancing numbers