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On the Hadamard Product of Balancing Matrix and Balancing Matrix

Yıl 2015, Cilt: 5 Sayı: 2, 201 - 207, 01.12.2015

Öz

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

Kaynakça

  • 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.
Yıl 2015, Cilt: 5 Sayı: 2, 201 - 207, 01.12.2015

Öz

Kaynakça

  • 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.
Toplam 23 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Research Article
Yazarlar

P. K. Ray Bu kişi benim

S. Swain Bu kişi benim

Yayımlanma Tarihi 1 Aralık 2015
Yayımlandığı Sayı Yıl 2015 Cilt: 5 Sayı: 2

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