[1] A. Faisant. On The Padovan Sequences. 2019. hal-02131654.
[2] A. D. Godase, M. B. Dhakne, On the Properties of k-Fibonacci and k-Lucas Numbers, Int. J. Adv.
Appl. Math. And Mech, 2014, pp. 100-106.
[3] T. He, J. H. Liao, P. J. Shiue, Matrix Representation of Recursive Sequences of Order 3 and Its
Applications, Journal of Mathematical Research with Applications ,May, 2018, pp. 221-235.
[4] G. C. Morales, New Identities for Padovan Sequences, http://orcid.org/0000-0003-3164-4434, 2019, pp.
1-9.
[5] S.H.J. Petroudi, M. Pirouz, On Pell -Narayana Sequences, 2nd national conference on mathematics and
statistics, Gonbad Kavous University, 2020, pp. 1-8.
[6] K. Sokhuma, Matrices Formula for Padovan and Perrin Sequences. Applied Mathematical Sciences 2013,
pp.7093-7096.
[7] F. Yilmaz, D. Bozkurt, Hessenberg Matrices and the Pell and Perrin Numbers. J. Number Theory,
2011, pp. 1390-1396.
[9] A. S. Liana, I. Wloch, Jacobsthal and Jacobsthal Lucas Hybrid Numbers, Annales Mathematiccae
Silesianae, 2019, pp. 276-283.
[10] A. Szynal-Liana. The Horadam Hybrid Numbers. Discuss Math Gen Algebra Appl. 2018;38(1): 91-98.
[11] T. D. Şentürk, G. Bilgici, A. Daşdemir, Z. Unal, Study on Horadam Hybrid Numbers,Turk.J. Math.
(2020) 44: 1212 - 1221.
[12] P. Catarino, G. Bilgici, A Note on Modied k-Pell Hybrid Numbers, Konuralp Journal of Mathematics,
8 (2) (2020) 229-233.
[13] S.H.J. Petroudi, M. Pirouz, On Narayana Hybrid Numbers, preprint, 2020.
[14] S.H.J. Petroudi, M. Pirouz, On Circulant Matrix Involving the Van Der Laan Hybrid Sequence, Preprint,
2nd National Conference on Mathematics and Statistics, Gonbad Kavous University, 2020, pp. 1-11.
[15] Emrah Polatlı, Hybrid Numbers with Fibonacci and Lucas Hybrid Number Coecients, preprint, 2020.
[16] A. Szynal-Liana, I. W loch, Introduction to Fibonacci and Lucas Hybrinomials, Complex Variables and
Elliptic Equations, 65:10, 1736-1747, 2020.
[17] C. Kızılateş, A Note on Horadam Hybrinomials, preprints, doi:10.20944/preprints202001.0116.v1, 2020.
[18] M. Liana, A. Szynal-Liana, I. W loch, On Pell hybrinomials, Miskolc Mathematical Notes, Vol. 20 (2019),
No. 2, pp. 1051-1062.
[19] A. Szynal-Liana, I. W loch, Generalized Fibonacci-Pell Hybrinomials, Online Journal of Analytic
Combinatorics, Issue 15, 1-12, (2020).
[20] J. P. Allouche, J. Johnson, Narayana's Cows and Delayed Morphisms, In: Articles of 3rd Computer
Music Conference JIM96, France, 1996.
[21] J. L. Ramirez, V. F. Sirvent, A Note on the k-Narayana Sequence, Annales Mathematicae et
Informaticae, 2015, pp. 91-105.
[22] G. Bilgici, The Generalized Order k-Narayana's Cows Numbers, Mathematica Slovaca 66(4) (2016),
795-802.
The Narayana Polynomial and Narayana Hybrinomial Sequences
Year 2021,
Volume: 9 Issue: 1, 90 - 99, 28.04.2021
Hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper we introduce the Narayana polynomial sequence(or polynomial sequence of Narayana's cows) and related Narayana hybrinomial sequence. We present Binet-like formula, generating function, exponential generating function of these sequences. In addition we give some identities such as Catalan-like identity, Cassini-like identity and Ocagne-like identity for these sequences.
[1] A. Faisant. On The Padovan Sequences. 2019. hal-02131654.
[2] A. D. Godase, M. B. Dhakne, On the Properties of k-Fibonacci and k-Lucas Numbers, Int. J. Adv.
Appl. Math. And Mech, 2014, pp. 100-106.
[3] T. He, J. H. Liao, P. J. Shiue, Matrix Representation of Recursive Sequences of Order 3 and Its
Applications, Journal of Mathematical Research with Applications ,May, 2018, pp. 221-235.
[4] G. C. Morales, New Identities for Padovan Sequences, http://orcid.org/0000-0003-3164-4434, 2019, pp.
1-9.
[5] S.H.J. Petroudi, M. Pirouz, On Pell -Narayana Sequences, 2nd national conference on mathematics and
statistics, Gonbad Kavous University, 2020, pp. 1-8.
[6] K. Sokhuma, Matrices Formula for Padovan and Perrin Sequences. Applied Mathematical Sciences 2013,
pp.7093-7096.
[7] F. Yilmaz, D. Bozkurt, Hessenberg Matrices and the Pell and Perrin Numbers. J. Number Theory,
2011, pp. 1390-1396.
[9] A. S. Liana, I. Wloch, Jacobsthal and Jacobsthal Lucas Hybrid Numbers, Annales Mathematiccae
Silesianae, 2019, pp. 276-283.
[10] A. Szynal-Liana. The Horadam Hybrid Numbers. Discuss Math Gen Algebra Appl. 2018;38(1): 91-98.
[11] T. D. Şentürk, G. Bilgici, A. Daşdemir, Z. Unal, Study on Horadam Hybrid Numbers,Turk.J. Math.
(2020) 44: 1212 - 1221.
[12] P. Catarino, G. Bilgici, A Note on Modied k-Pell Hybrid Numbers, Konuralp Journal of Mathematics,
8 (2) (2020) 229-233.
[13] S.H.J. Petroudi, M. Pirouz, On Narayana Hybrid Numbers, preprint, 2020.
[14] S.H.J. Petroudi, M. Pirouz, On Circulant Matrix Involving the Van Der Laan Hybrid Sequence, Preprint,
2nd National Conference on Mathematics and Statistics, Gonbad Kavous University, 2020, pp. 1-11.
[15] Emrah Polatlı, Hybrid Numbers with Fibonacci and Lucas Hybrid Number Coecients, preprint, 2020.
[16] A. Szynal-Liana, I. W loch, Introduction to Fibonacci and Lucas Hybrinomials, Complex Variables and
Elliptic Equations, 65:10, 1736-1747, 2020.
[17] C. Kızılateş, A Note on Horadam Hybrinomials, preprints, doi:10.20944/preprints202001.0116.v1, 2020.
[18] M. Liana, A. Szynal-Liana, I. W loch, On Pell hybrinomials, Miskolc Mathematical Notes, Vol. 20 (2019),
No. 2, pp. 1051-1062.
[19] A. Szynal-Liana, I. W loch, Generalized Fibonacci-Pell Hybrinomials, Online Journal of Analytic
Combinatorics, Issue 15, 1-12, (2020).
[20] J. P. Allouche, J. Johnson, Narayana's Cows and Delayed Morphisms, In: Articles of 3rd Computer
Music Conference JIM96, France, 1996.
[21] J. L. Ramirez, V. F. Sirvent, A Note on the k-Narayana Sequence, Annales Mathematicae et
Informaticae, 2015, pp. 91-105.
[22] G. Bilgici, The Generalized Order k-Narayana's Cows Numbers, Mathematica Slovaca 66(4) (2016),
795-802.
Petroudi, S., Pirouz, M., & Özkoç, A. (2021). The Narayana Polynomial and Narayana Hybrinomial Sequences. Konuralp Journal of Mathematics, 9(1), 90-99.
AMA
Petroudi S, Pirouz M, Özkoç A. The Narayana Polynomial and Narayana Hybrinomial Sequences. Konuralp J. Math. April 2021;9(1):90-99.
Chicago
Petroudi, Seyyed, Maryam Pirouz, and Arzu Özkoç. “The Narayana Polynomial and Narayana Hybrinomial Sequences”. Konuralp Journal of Mathematics 9, no. 1 (April 2021): 90-99.
EndNote
Petroudi S, Pirouz M, Özkoç A (April 1, 2021) The Narayana Polynomial and Narayana Hybrinomial Sequences. Konuralp Journal of Mathematics 9 1 90–99.
IEEE
S. Petroudi, M. Pirouz, and A. Özkoç, “The Narayana Polynomial and Narayana Hybrinomial Sequences”, Konuralp J. Math., vol. 9, no. 1, pp. 90–99, 2021.
ISNAD
Petroudi, Seyyed et al. “The Narayana Polynomial and Narayana Hybrinomial Sequences”. Konuralp Journal of Mathematics 9/1 (April 2021), 90-99.
JAMA
Petroudi S, Pirouz M, Özkoç A. The Narayana Polynomial and Narayana Hybrinomial Sequences. Konuralp J. Math. 2021;9:90–99.
MLA
Petroudi, Seyyed et al. “The Narayana Polynomial and Narayana Hybrinomial Sequences”. Konuralp Journal of Mathematics, vol. 9, no. 1, 2021, pp. 90-99.
Vancouver
Petroudi S, Pirouz M, Özkoç A. The Narayana Polynomial and Narayana Hybrinomial Sequences. Konuralp J. Math. 2021;9(1):90-9.