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On the Hadamard Product of Balancing Matrix and Balancing Matrix

Year 2015, Volume: 5 Issue: 2, 201 - 207, 01.12.2015

Abstract

In this paper, the matrix Q n B ◦ Q −n B which is the Hadamard product of both balancing Q n B matrix and balancing Q −n B matrix is introduced. Some properties of the Hadamard product of these matrices are investigated. A different coding and decoding method based on the application of the Hadamard product of balancing Q n B matrix and balancing Q −n B matrix is also considered

References

  • Behera, A. and Panda, G.K., (1999), On the square roots of triangular numbers, 37(2), Fibonacci Quart., pp. 98-105.
  • B`erczes, A., Liptai, K. and Pink, I., (2010), On generalized balancing numbers, Fibonacci Quart., (2), pp. 121-128.
  • Horn, R.A. and Johnson, C.A., (1985), Matrix Analysis, Cambridge University Press, New York.
  • Keskin R. and Karaatly O., (2012), Some new properties of balancing numbers and square triangular numbers, J.Integer Seq., 15(1), pp. 12.1.4.
  • Liptai, K., Fibonacci balancing numbers, (2004), Fibonacci Quart., 42(4), pp. 330-340.
  • Liptai, K., Lucas balancing numbers, (2006), Acta Math. Univ. Ostrav., 14(1), pp. 43-47.
  • Liptai, K., Luca, F., Pinter A. and Szalay L. , (2009), Generalized balancing numbers, Indag. Math.(N. S.), 20, pp. 87-100.
  • Olajos, P., (2010), Properties of balancing, cobalancing and generalized balancing numbers, Ann. Math. Inform. , 37, pp. 125-138.
  • Panda, G.K. and Ray P.K., (2011), Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bull. Inst. Math. Acad. Sin. (N. S.), 6(1), pp. 41-72.
  • Panda, G.K. and Ray P.K., (2005), Cobalancing numbers and cobalancers, Int. J. of Math. Math. Sci., 8, pp. 1189-1200.
  • Panda, G.K. (2009), Some fascinating properties of balancing numbers, Proceeding of the Eleventh
  • International Conference on Fibonacci Numbers and Their Applications, Congr. Numer. , 194, pp. 189. Patel, B.K. and Ray, P.K. (2015), The Period, rank and order of the sequence of balancing numbers modulo m, accepted in Mathematical Reports.
  • Ray P.K., (2012), Application of Chybeshev polynomials in factorization of balancing and Lucas- balancing numbers, Bol, Soc. Parana. Mat. , 30 (2), pp. 49-56.
  • Ray P.K., (2012), Certain matrices associated with balancing and Lucas-balancing numbers, Matem- atika, 28 (1), pp. 15-22.
  • Ray P.K., (2013), Factorization of negatively subscripted balancing and Lucas-balancing numbers
  • Bol, Soc. Parana. Mat. , 31 (2), pp. 161-173. Ray P.K., (2012), Curious congruences for balancing numbers, Int. J. of Contemp. Math. Sci., 7 (18), pp. 881-889.
  • Ray P.K., (2013), New identities for the common factors for balancing and Lucas-balancing numbers
  • Int. J. Pure Appl. Math., 85, pp. 487-494. Ray P.K., (2014), Some congruences for balancing and Lucas-balancing numbers and their applica- tions, Integers, 14, #A8.
  • Ray P.K., (2014), On the properties of Lucas-balancing numbers by matrix method, Sigmae, Alfenas, (1), pp. 1-6.
  • Ray P.K., Parida K., (2014), Generalization of Cassini formula for balancing and Lucas-balancing numbers, Matematychni Studii., 42(1), pp. 9-14.
  • Ray P.K., Dila G.K., Patel B.K., (2014), Application of some recurrence relations to cryptography using finite state machine International Journal of Computer Science and Electronics Engineering (IJCSEE), 2 (4), pp. 220-223.
  • Ray P.K., (2014), Identities involving the terms of a balancing-like sequence via matrices, Caspian
  • Journal of Applied Mathematics, Ecology and Economics, 2(1), pp. 94-100. Ray P.K., (2015), Balancing and Lucas balancing sums by matrix methods, Mathematical Reports, (67), 2, pp. 225-233.
Year 2015, Volume: 5 Issue: 2, 201 - 207, 01.12.2015

Abstract

References

  • Behera, A. and Panda, G.K., (1999), On the square roots of triangular numbers, 37(2), Fibonacci Quart., pp. 98-105.
  • B`erczes, A., Liptai, K. and Pink, I., (2010), On generalized balancing numbers, Fibonacci Quart., (2), pp. 121-128.
  • Horn, R.A. and Johnson, C.A., (1985), Matrix Analysis, Cambridge University Press, New York.
  • Keskin R. and Karaatly O., (2012), Some new properties of balancing numbers and square triangular numbers, J.Integer Seq., 15(1), pp. 12.1.4.
  • Liptai, K., Fibonacci balancing numbers, (2004), Fibonacci Quart., 42(4), pp. 330-340.
  • Liptai, K., Lucas balancing numbers, (2006), Acta Math. Univ. Ostrav., 14(1), pp. 43-47.
  • Liptai, K., Luca, F., Pinter A. and Szalay L. , (2009), Generalized balancing numbers, Indag. Math.(N. S.), 20, pp. 87-100.
  • Olajos, P., (2010), Properties of balancing, cobalancing and generalized balancing numbers, Ann. Math. Inform. , 37, pp. 125-138.
  • Panda, G.K. and Ray P.K., (2011), Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bull. Inst. Math. Acad. Sin. (N. S.), 6(1), pp. 41-72.
  • Panda, G.K. and Ray P.K., (2005), Cobalancing numbers and cobalancers, Int. J. of Math. Math. Sci., 8, pp. 1189-1200.
  • Panda, G.K. (2009), Some fascinating properties of balancing numbers, Proceeding of the Eleventh
  • International Conference on Fibonacci Numbers and Their Applications, Congr. Numer. , 194, pp. 189. Patel, B.K. and Ray, P.K. (2015), The Period, rank and order of the sequence of balancing numbers modulo m, accepted in Mathematical Reports.
  • Ray P.K., (2012), Application of Chybeshev polynomials in factorization of balancing and Lucas- balancing numbers, Bol, Soc. Parana. Mat. , 30 (2), pp. 49-56.
  • Ray P.K., (2012), Certain matrices associated with balancing and Lucas-balancing numbers, Matem- atika, 28 (1), pp. 15-22.
  • Ray P.K., (2013), Factorization of negatively subscripted balancing and Lucas-balancing numbers
  • Bol, Soc. Parana. Mat. , 31 (2), pp. 161-173. Ray P.K., (2012), Curious congruences for balancing numbers, Int. J. of Contemp. Math. Sci., 7 (18), pp. 881-889.
  • Ray P.K., (2013), New identities for the common factors for balancing and Lucas-balancing numbers
  • Int. J. Pure Appl. Math., 85, pp. 487-494. Ray P.K., (2014), Some congruences for balancing and Lucas-balancing numbers and their applica- tions, Integers, 14, #A8.
  • Ray P.K., (2014), On the properties of Lucas-balancing numbers by matrix method, Sigmae, Alfenas, (1), pp. 1-6.
  • Ray P.K., Parida K., (2014), Generalization of Cassini formula for balancing and Lucas-balancing numbers, Matematychni Studii., 42(1), pp. 9-14.
  • Ray P.K., Dila G.K., Patel B.K., (2014), Application of some recurrence relations to cryptography using finite state machine International Journal of Computer Science and Electronics Engineering (IJCSEE), 2 (4), pp. 220-223.
  • Ray P.K., (2014), Identities involving the terms of a balancing-like sequence via matrices, Caspian
  • Journal of Applied Mathematics, Ecology and Economics, 2(1), pp. 94-100. Ray P.K., (2015), Balancing and Lucas balancing sums by matrix methods, Mathematical Reports, (67), 2, pp. 225-233.
There are 23 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

P. K. Ray This is me

S. Swain This is me

Publication Date December 1, 2015
Published in Issue Year 2015 Volume: 5 Issue: 2

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