Araştırma Makalesi
BibTex RIS Kaynak Göster

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

Yıl 2017, Cilt: 46 Sayı: 3, 361 - 372, 01.06.2017

Paula Catarino Helena Campos

Öz

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.

Anahtar Kelimeler

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

Kaynakça

  • 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.

Yıl 2017, Cilt: 46 Sayı: 3, 361 - 372, 01.06.2017

Paula Catarino Helena Campos

Öz

Kaynakça

  • 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.
Toplam 35 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Paula Catarino

Helena Campos Bu kişi benim

Yayımlanma Tarihi 1 Haziran 2017
Yayımlandığı Sayı Yıl 2017 Cilt: 46 Sayı: 3

Kaynak Göster

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. Haziran 2017;46(3):361-372.
Chicago Catarino, Paula, ve Helena Campos. “Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell Numbers”. Hacettepe Journal of Mathematics and Statistics 46, sy. 3 (Haziran 2017): 361-72.
EndNote Catarino P, Campos H (01 Haziran 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 ve H. Campos, “Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell numbers”, Hacettepe Journal of Mathematics and Statistics, c. 46, sy. 3, ss. 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 (Haziran 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 ve Helena Campos. “Incomplete $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell Numbers”. Hacettepe Journal of Mathematics and Statistics, c. 46, sy. 3, 2017, ss. 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