Notes on Binomial Transform of the Generalized Narayana Sequence

  • Yüksel Soykan Department of Mathematics, Art and Science Faculty, Zonguldak Bülent Ecevit University, 67100, Zonguldak, Turkey
Keywords: binomial transform, Narayana sequence, Narayana numbers, binomial transform of Narayana sequence, generalized Tribonacci sequence

Abstract

In this paper, we define the binomial transform of the generalized Narayana sequence and as special cases, the binomial transform of the Narayana, Narayana-Lucas, Narayana-Perrin sequences will be introduced. We investigate their properties in details.

References

P. Barry, On integer-sequence-based constructions of generalized Pascal triangles, Journal of Integer Sequences 9 (2006), Article 06.2.4.

I. Bruce, A modified Tribonacci sequence, The Fibonacci Quarterly 22(3) (1984), 244-246.

M. Catalani, Identities for Tribonacci-related sequences, 2002. arXiv preprint: https://arxiv.org/pdf/math/0209179.pdf math/0209179

E. Choi, Modular Tribonacci numbers by matrix method, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 20(3) (2013), 207-221. https://doi.org/10.7468/jksmeb.2013.20.3.207

M. Elia, Derived sequences, the Tribonacci recurrence and cubic forms, The Fibonacci Quarterly 39(2) (2001), 107-115.

M. C. Er, Sums of Fibonacci numbers by matrix methods, Fibonacci Quart. 22(3) (1984), 204-207.

H. W. Gould, Series transformations for finding recurrences for sequences, The Fibonacci Quarterly 28(2) (1990), 166-171.

P. Haukkanen, Formal power series for binomial sums of sequences of numbers, The Fibonacci Quarterly 31(1) (1993), 28-31.

A. F. Horadam, A generalized Fibonacci sequence, American Mathematical Monthly 68 (1961), 455-459. https://doi.org/10.1080/00029890.1961.11989696

A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quarterly 3.3 (1965), 161-176.

A. F. Horadam, Special properties of the sequence $w_{n}(a,b;p,q)$, Fibonacci Quarterly 5(5) (1967), 424-434.

A. F. Horadam, Generating functions for powers of a certain generalized sequence of numbers, Duke Math. J. 32 (1965), 437-446. https://doi.org/10.1215/S0012-7094-65-03244-8

F.T. Howard and F. Saidak, Zhou's theory of constructing identities, Congress Numer. 200 (2010), 225-237.

D. Kalman, Generalized Fibonacci numbers by matrix methods, Fibonacci Quart. 20(1) (1982), 73-76.

D. E. Knuth, The Art of Computer Programming 3, Reading, MA: Addison Wesley, 1973.

P. Y. Lin, De Moivre-Type identities for the Tribonacci numbers, The Fibonacci Quarterly 26 (1988), 131-134.

S. Pethe, Some identities for Tribonacci sequences, The Fibonacci Quarterly 26(2) (1988), 144-151.

H. Prodinger, Some information about the Binomial transform, The Fibonacci Quarterly 32.5 (1994), 412-415.

A. Scott, T. Delaney and V. Hoggatt, Jr., The Tribonacci sequence, The Fibonacci Quarterly 15(3) (1977), 193-200.

N. J. A. Sloane, The On-line Encyclopedia of Integer Sequences, http://oeis.org/

A. Shannon, Tribonacci numbers and Pascal's pyramid, The Fibonacci Quarterly 15(3) (1977), pp. 268 and 275.

Y. Soykan, Simson identity of generalized m-step Fibonacci numbers, Int. J. Adv. Appl. Math. Mech. 7(2) (2019), 45-56.

Y. Soykan, Tribonacci and Tribonacci-Lucas sedenions, Mathematics 7(1) (2019), 74. https://doi.org/10.3390/math7010074

Y. Soykan, Summing formulas for generalized Tribonacci numbers, Universal Journal of Mathematics and Applications 3(1) (2020), 1-11. https://doi.org/10.32323/ujma.637876

Y. Soykan, Generalized Tribonacci numbers: summing formulas, Int. J. Adv. Appl. Math. Mech. 7(3) (2020), 57-76.

Y. Soykan, A closed formula for the sums of squares of generalized Tribonacci numbers, Journal of Progressive Research in Mathematics 16(2) (2020), 2932-2941.

Y. Soykan, On the sums of squares of generalized Tribonacci numbers: closed formulas of $sum_{k=0}^{n}x^{k}W_{k}^{2}$, Archives of Current Research International 20(4) (2020), 22-47. https://doi.org/10.9734/acri/2020/v20i430187

Y. Soykan, On generalized Narayana numbers, Int. J. Adv. Appl. Math. Mech. 7(3) (2020), 43-56.

W. Spickerman, Binet's formula for the Tribonacci sequence, The Fibonacci Quarterly 20 (1982), 118-120.

M. Z. Spivey, Combinatorial sums and finite differences, Discrete Math. 307 (2007), 3130-3146. https://doi.org/10.1016/j.disc.2007.03.052

C. C. Yalavigi, Properties of Tribonacci numbers, The Fibonacci Quarterly 10(3) (1972), 231-246.

Published
2021-06-02
How to Cite
Soykan, Y. (2021). Notes on Binomial Transform of the Generalized Narayana Sequence. Earthline Journal of Mathematical Sciences, 7(1), 77-111. https://doi.org/10.34198/ejms.7121.77111
Section
Articles