Research Article
BibTex RIS Cite

Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell numbers

Year 2017, Volume: 46 Issue: 3, 361 - 372, 01.06.2017

Paula Catarino , Helena Campos

Abstract

In this paper, it is defined the incomplete $k$-Pell, $k$-Pell-Lucas and Modified $k$-Pell numbers, it is studied the recurrence relations, some properties of these sequences of integers and their generating functions.

Keywords

Incomplete Pell numbers Incomplete Pell-Lucas numbers Incomplete Modified Pell numbers $k$-Pell; $k$-Pell-Lucas and modified $k$-Pell numbers

References

  • Aires, A. P., Catarino, P., Campos, H., Borges, A. and Vasco, P. $s$-Generalized Fibonacci Numbers: Some Identities, a Generating Function and Pythagorean Triples, Int. J. Math. Anal. 8 (36), 1757-1766, 2014.
  • Bolat, C. and Köse, H. On the Properties of $k$-Fibonacci Numbers, Int. J. Contemp. Math. Sci. 22 (5), 1097-1105, 2010.
  • Catarino, P. On some identities for $k$-Fibonacci sequence, Int. J. Contemp. Math. Sci. 9 (1), 37-42, 2014.
  • Catarino, P. A note involving Two-by-Two Matrices of the $k$-Pell and $k$-Pell-Lucas Se- quences, Int. Math. Forum 8 (32), 1561-1568, 2013.
  • Catarino, P. On some identities and generating functions for $k$-Pell numbers, Int. J. Math. Anal. 7 (38), 1877-1884, 2013.
  • Catarino, P. and Vasco, P. On Some Identities and Generating Functions for $k$-Pell-Lucas Sequence, Appl. Math. Sci. 7 (98), 4867-4873, 2013.
  • Catarino, P. and Vasco, P. Modified $k$-Pell Sequence: Some Identities and Ordinary Gen- erating Function, Appl. Math. Sci. 7 (121), 6031-6037, 2013.
  • Catarino, P., Vasco, P., Borges, A., Campos, H., and Aires, A. P. Sums, Products and identities involving $k$-Fibonacci and $k$-Lucas sequences, JP J. Algebra Number Theory Appl. 32 (1), 63-77, 2014.
  • Djordjevi¢, G. B. Convolutions of the Generalized Pell and Pell-Lucas numbers, Filomat 30 (1), 105-112, 2016.
  • Djordjevi¢, G. B. Generating functions of the incomplete generalized Fibonacci and gener- alized Lucas numbers, Fibonacci Quart. 42 (2), 106-113, 2004.
  • Djordjevi¢, G. B. and Srivastava, H. M. Incomplete generalized Jacobsthal and Jacobsthal- Lucas numbers, Math. Comput. Modelling 42 (9-10), 1049-1056, 2005.
  • Falcon, S. On the Lucas Triangle and its Relationship with the $k$Lucas numbers, J. Math. Comput. Sci. 2 (3), 425-434, 2012.
  • Falcon, S. On the $k$Lucas numbers, Int. J. Contemp. Math. Sci. 6 (21), 1039-1050, 2011.
  • Falcon, S. and Plaza, A. On the Fibonacci $k$numbers, Chaos, Solitons Fractals 32 (5), 1615-1624, 2007.
  • Filipponi, P. Incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo 45 (2), 3756, 1996.
  • Gupta, V. K., Panwar, Y. K. and Sikhwal, O. Generalized Fibonacci Sequences, Theoretical Mathematics and Applications 2 (2), 115-124, 2012.
  • Halici, S. Some sums formulae for products of terms of Pell, Pell-Lucas and Modified Pell sequences, SAÜ. Fen Bilimleri Dergisi 15 Cilt, 2. SayL, 151-155, 2011.
  • Hoggatt, V. E. Fibonacci and Lucas Numbers, Fibonacci Association, University of Santa Clara, Santa Clara, Houghton Miin Company, 1969.
  • Horadam, A. F. Pell identities, Fibonacci Quart. 9 (3), 245-252, 263, 1971.
  • Koshy, T. Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.
  • Koshy, T. Fibonacci, Lucas, and Pell numbers, and Pascal's triangle, Mathematical Spectrum 43 (3), 125-132, 2011.
  • Luca, F. Fibonacci numbers, Lucas numbers and orders of nite simple groups, JP J. Algebra Number Theory Appl. 4 (1), 23-54, 2004.
  • Marques, D. The Order of Appearance of the Product of Consecutive Lucas Numbers, Fibonacci Quart. 51 (1), 38-43, 2013.
  • Pintér, A. and Srivastava, H. M. Generating functions of the incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo 48 (Serie II), 591-596, 1999.
  • Ramírez, J. L. Incomplete generalized Fibonacci and Lucas polynomials, Hacettepe Journal of Mathematics and Statistics 44 (2), 363-373, 2015.
  • Ramírez, J. and Sirvent, V. Incomplete tribonacci numbers and polynomials, J. Integer Seq. 17, article 14.4.2, 2014.
  • Srivastava, H. M. and Manocha, H. L. A Treatise on Generating Functions, Halsted Press, Ellis Horwood Limited, Chichester, UK, John Wiley and Sons, New York, NY, USA, 1984.
  • Tasci, D., Cetin Firengiz, M. and Tuglu, N. Incomplete bivariate Fibonacci and Lucas ppoly- nomials, Discrete Dyn. Nat. Soc., article ID 840345, 2012.
  • Tasci, D. and Firengiz, M. C. Incomplete Fibonacci and Lucas $p$-numbers, Math. Comput. Modelling 52 (9-10), 1763-1770, 2010.
  • Vasco, P., Catarino, P., Campos, H., Aires, A. P. and Borges, A. $k$-Pell, $k$-Pell-Lucas and Modied $k$-Pell Numbers: Some Identities and Norms of Hankel Matrices, Int. J. Math. Anal. 9 (1), 31-37, 2015.
  • Vasco, P. and Catarino, P. Sums and products involving terms of $k$-Pell, $k$-Pell-Lucas and Modified $k$-Pell sequences, JP J. Algebra Number Theory Appl. 32 (2), 87-98, (2014).
  • Vorobiov, N. N. Numeros de Fibonacci, Editora MIR, URSS, 1974.
  • Yilmaz, N. Incomplete Tribonacci Numbers and Its Determinants, Ms.Thesis, Selçuk University, 2011.
  • Yilmaz, N. and Taskara, N. Incomplete Tribonacci-Lucas Numbers and Polynomials, Adv. Appl. Cliord Algebras 25, 741-753, 2015.
  • Young H. J. and Sang P. J. Linearlization of Generalized Fibonacci Sequences, Korean J. Math. 22 (3), 443-454, 2014.

Year 2017, Volume: 46 Issue: 3, 361 - 372, 01.06.2017

Paula Catarino , Helena Campos

Abstract

References

  • Aires, A. P., Catarino, P., Campos, H., Borges, A. and Vasco, P. $s$-Generalized Fibonacci Numbers: Some Identities, a Generating Function and Pythagorean Triples, Int. J. Math. Anal. 8 (36), 1757-1766, 2014.
  • Bolat, C. and Köse, H. On the Properties of $k$-Fibonacci Numbers, Int. J. Contemp. Math. Sci. 22 (5), 1097-1105, 2010.
  • Catarino, P. On some identities for $k$-Fibonacci sequence, Int. J. Contemp. Math. Sci. 9 (1), 37-42, 2014.
  • Catarino, P. A note involving Two-by-Two Matrices of the $k$-Pell and $k$-Pell-Lucas Se- quences, Int. Math. Forum 8 (32), 1561-1568, 2013.
  • Catarino, P. On some identities and generating functions for $k$-Pell numbers, Int. J. Math. Anal. 7 (38), 1877-1884, 2013.
  • Catarino, P. and Vasco, P. On Some Identities and Generating Functions for $k$-Pell-Lucas Sequence, Appl. Math. Sci. 7 (98), 4867-4873, 2013.
  • Catarino, P. and Vasco, P. Modified $k$-Pell Sequence: Some Identities and Ordinary Gen- erating Function, Appl. Math. Sci. 7 (121), 6031-6037, 2013.
  • Catarino, P., Vasco, P., Borges, A., Campos, H., and Aires, A. P. Sums, Products and identities involving $k$-Fibonacci and $k$-Lucas sequences, JP J. Algebra Number Theory Appl. 32 (1), 63-77, 2014.
  • Djordjevi¢, G. B. Convolutions of the Generalized Pell and Pell-Lucas numbers, Filomat 30 (1), 105-112, 2016.
  • Djordjevi¢, G. B. Generating functions of the incomplete generalized Fibonacci and gener- alized Lucas numbers, Fibonacci Quart. 42 (2), 106-113, 2004.
  • Djordjevi¢, G. B. and Srivastava, H. M. Incomplete generalized Jacobsthal and Jacobsthal- Lucas numbers, Math. Comput. Modelling 42 (9-10), 1049-1056, 2005.
  • Falcon, S. On the Lucas Triangle and its Relationship with the $k$Lucas numbers, J. Math. Comput. Sci. 2 (3), 425-434, 2012.
  • Falcon, S. On the $k$Lucas numbers, Int. J. Contemp. Math. Sci. 6 (21), 1039-1050, 2011.
  • Falcon, S. and Plaza, A. On the Fibonacci $k$numbers, Chaos, Solitons Fractals 32 (5), 1615-1624, 2007.
  • Filipponi, P. Incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo 45 (2), 3756, 1996.
  • Gupta, V. K., Panwar, Y. K. and Sikhwal, O. Generalized Fibonacci Sequences, Theoretical Mathematics and Applications 2 (2), 115-124, 2012.
  • Halici, S. Some sums formulae for products of terms of Pell, Pell-Lucas and Modified Pell sequences, SAÜ. Fen Bilimleri Dergisi 15 Cilt, 2. SayL, 151-155, 2011.
  • Hoggatt, V. E. Fibonacci and Lucas Numbers, Fibonacci Association, University of Santa Clara, Santa Clara, Houghton Miin Company, 1969.
  • Horadam, A. F. Pell identities, Fibonacci Quart. 9 (3), 245-252, 263, 1971.
  • Koshy, T. Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.
  • Koshy, T. Fibonacci, Lucas, and Pell numbers, and Pascal's triangle, Mathematical Spectrum 43 (3), 125-132, 2011.
  • Luca, F. Fibonacci numbers, Lucas numbers and orders of nite simple groups, JP J. Algebra Number Theory Appl. 4 (1), 23-54, 2004.
  • Marques, D. The Order of Appearance of the Product of Consecutive Lucas Numbers, Fibonacci Quart. 51 (1), 38-43, 2013.
  • Pintér, A. and Srivastava, H. M. Generating functions of the incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo 48 (Serie II), 591-596, 1999.
  • Ramírez, J. L. Incomplete generalized Fibonacci and Lucas polynomials, Hacettepe Journal of Mathematics and Statistics 44 (2), 363-373, 2015.
  • Ramírez, J. and Sirvent, V. Incomplete tribonacci numbers and polynomials, J. Integer Seq. 17, article 14.4.2, 2014.
  • Srivastava, H. M. and Manocha, H. L. A Treatise on Generating Functions, Halsted Press, Ellis Horwood Limited, Chichester, UK, John Wiley and Sons, New York, NY, USA, 1984.
  • Tasci, D., Cetin Firengiz, M. and Tuglu, N. Incomplete bivariate Fibonacci and Lucas ppoly- nomials, Discrete Dyn. Nat. Soc., article ID 840345, 2012.
  • Tasci, D. and Firengiz, M. C. Incomplete Fibonacci and Lucas $p$-numbers, Math. Comput. Modelling 52 (9-10), 1763-1770, 2010.
  • Vasco, P., Catarino, P., Campos, H., Aires, A. P. and Borges, A. $k$-Pell, $k$-Pell-Lucas and Modied $k$-Pell Numbers: Some Identities and Norms of Hankel Matrices, Int. J. Math. Anal. 9 (1), 31-37, 2015.
  • Vasco, P. and Catarino, P. Sums and products involving terms of $k$-Pell, $k$-Pell-Lucas and Modified $k$-Pell sequences, JP J. Algebra Number Theory Appl. 32 (2), 87-98, (2014).
  • Vorobiov, N. N. Numeros de Fibonacci, Editora MIR, URSS, 1974.
  • Yilmaz, N. Incomplete Tribonacci Numbers and Its Determinants, Ms.Thesis, Selçuk University, 2011.
  • Yilmaz, N. and Taskara, N. Incomplete Tribonacci-Lucas Numbers and Polynomials, Adv. Appl. Cliord Algebras 25, 741-753, 2015.
  • Young H. J. and Sang P. J. Linearlization of Generalized Fibonacci Sequences, Korean J. Math. 22 (3), 443-454, 2014.
There are 35 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Paula Catarino

Helena Campos This is me

Publication Date June 1, 2017
Published in Issue Year 2017 Volume: 46 Issue: 3

Cite

APA Catarino, P., & Campos, H. (2017). Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell numbers. Hacettepe Journal of Mathematics and Statistics, 46(3), 361-372.
AMA Catarino P, Campos H. Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell numbers. Hacettepe Journal of Mathematics and Statistics. June 2017;46(3):361-372.
Chicago Catarino, Paula, and Helena Campos. “Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell Numbers”. Hacettepe Journal of Mathematics and Statistics 46, no. 3 (June 2017): 361-72.
EndNote Catarino P, Campos H (June 1, 2017) Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell numbers. Hacettepe Journal of Mathematics and Statistics 46 3 361–372.
IEEE P. Catarino and H. Campos, “Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell numbers”, Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 3, pp. 361–372, 2017.
ISNAD Catarino, Paula - Campos, Helena. “Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell Numbers”. Hacettepe Journal of Mathematics and Statistics 46/3 (June 2017), 361-372.
JAMA Catarino P, Campos H. Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell numbers. Hacettepe Journal of Mathematics and Statistics. 2017;46:361–372.
MLA Catarino, Paula and Helena Campos. “Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell Numbers”. Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 3, 2017, pp. 361-72.
Vancouver Catarino P, Campos H. Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell numbers. Hacettepe Journal of Mathematics and Statistics. 2017;46(3):361-72.

https://dergipark.org.tr/en/pub/hujms