Matrix Manipulations for Properties of Pell p-Numbers and their Generalizations

Erdağ, Özgür; Deveci, Ömür; Shannon, Anthony G.

doi:10.2478/auom-2020-0036

Abstract

In this paper, we define the Pell-Pell p-sequence and then we discuss the connection of the Pell-Pell p-sequence with Pell and Pell p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Pell-Pell p-numbers by the aid of the nth power of the generating matrix the Pell-Pell p-sequence. Furthermore, we obtain an exponential representation of the Pell-Pell p-numbers and we develop relationships between the Pell-Pell p-numbers and their permanent, determinant and sums of certain matrices.

References

[1] Y. Akuzum, O. Deveci and A. G. Shannon, On the Pell p-circulant Sequences, Notes on Number Theory and Disc. Math., 23 (2017), no. 2, 91-103.
Search in Google Scholar Back to article
[2] B. Bradie, Extension and refinements of some properties of sums involving Pell number, Missouri J.Math. Sci., 22 (2010), no. 1, 37-43.
Search in Google Scholar Back to article
[3] R. A. Brualdi and P. M. Gibson, Convex polyhedra of doubly stochastic matrices I: applications of permanent function, J. Combin. Theory, Series A, 22 (1997), no. 2, 194-230.
Search in Google Scholar Back to article
[4] W. Y. C. Chen and J. D. Louck, The combinatorial power of the companion matrix, Linear Algebra Appl., 232 (1996), 261-278.10.1016/0024-3795(95)90163-9
Search in Google Scholar Back to article
[5] R. Devaney, The Mandelbrot set and the Farey tree, and the Fibonacci sequence, Amer. Math. Monthly, 106 (1999), 289-302.10.1080/00029890.1999.12005046
Search in Google Scholar Back to article
[6] O. Deveci, The Pell-Circulant Sequences and Their Applications, Maejo Int. J. Sci. Technol., 10 (2016), no. 03, 284-293.
Search in Google Scholar Back to article
[7] O. Deveci and E. Karaduman, On the Padovan p-numbers, Hacettepe J. Math. Stat., 46 (2017), no. 4, 579-592.
Search in Google Scholar Back to article
[8] D. D. Frey and J. A. Sellers, Jacobsthal numbers and alternating sign matrices, J. Integer Seq., 3 (2000), Article 00.2.3.
Search in Google Scholar Back to article
[9] N. Gogin and A. A. Myllari, The Fibonacci-Padovan sequence and MacWilliams transform matrices, Programing and Computer Software, published in Programmirovanie, 33 (2007), no. 2, 74-79.
Search in Google Scholar Back to article
[10] A. F. Horadam, Jacobsthal representations numbers, Fibonacci Quart., 34 (1996), 40-54.
Search in Google Scholar Back to article
[11] A. F. Horadam, Applications of modified Pell numbers to representations, Ulam Quart., 3 (1994), no. 1, 34-53.
Search in Google Scholar Back to article
[12] B. Johnson, Fibonacci identities by matrix methods and generalisation to related sequences, http://maths.dur.ac.uk/~dma0rcj/PED/fib.pdf, March 25, 2003.
Search in Google Scholar Back to article
[13] D. Kalman, Generalized Fibonacci numbers by matrix methods, Fibonacci Quart., 20 (1982), no. 1, 73-76.
Search in Google Scholar Back to article
[14] E. Kilic, The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial representations, sums, Chaos, Solitons Fractals, 40 (2009), no. 4, 2047-2063.
Search in Google Scholar Back to article
[15] E. Kilic, The Binet fomula, sums and representations of generalized Fibonacci p-numbers, Eur. J. Comb., 29 (2008), 701–711.10.1016/j.ejc.2007.03.004
Search in Google Scholar Back to article
[16] E. Kilic and D.Tasci, The generalized Binet formula, representation and sums of the generalized order-k Pell numbers, Taiwanese J. Math., 10 (2006), no. 6, 1661-1670.
Search in Google Scholar Back to article
[17] E. G. Kocer, The Binet formulas for the Pell and Pell-Lucas p-numbers, Ars Comb., 85 (2007), 3-17.
Search in Google Scholar Back to article
[18] F. Koken and D. Bozkurt, On the Jacobsthal numbers by matrix methods, Int. J. Contemp. Math. Sciences. 3 (2008), no. 13, 605-614.
Search in Google Scholar Back to article
[19] P. Lancaster and M. Tismenetsky, The theory of matrices: with applications, Elsevier, 1985.
Search in Google Scholar Back to article
[20] R. Lidl and H. Niederreiter, Introduction to finite fields and their applications, Cambridge UP, 1986.
Search in Google Scholar Back to article
[21] W. L. McDaniel, Triangular numbers in the Pell sequence, Fibonacci Quart., 34 (1996), 105-107.
Search in Google Scholar Back to article
[22] A. G. Shannon, P. G. Anderson and A. F. Horadam, Properties of cordonnier Perrin and Van der Lan numbers, Internat. J. Math. Ed. Sci. Tech., 37 (2006), no. 7, 825-831.
Search in Google Scholar Back to article
[23] A. G. Shannon, A. F. Horadam and P. G. Anderson, The auxiliary equation associated with plastic number, Notes on Number Theory and Disc. Math., 12 (2006), no. 1, 1-12.
Search in Google Scholar Back to article
[24] A.P. Stakhov, A generalization of the Fibonacci Q-matrix, Rep. Natl. Acad. Sci. Ukraine, 9 (1999), 46-49.
Search in Google Scholar Back to article
[25] A. P. Stakhov, B. Rozin, Theory of Binet formulas for Fibonacci and Lucas p-numbers, Chaos, Solitions Fractals, 27 (2006), 1162-77.10.1016/j.chaos.2005.04.106
Search in Google Scholar Back to article
[26] I. Stewart, Tales of a neglected number, Sci. Amer. 274 (1996), 102-103.
Search in Google Scholar Back to article
[27] D. Tasci and M. C. Firengiz, Incomplete Fibonacci and Lucas p-numbers, Math. Comput. Modell., 52 (2010), 1763-1770.10.1016/j.mcm.2010.07.003
Search in Google Scholar Back to article

Matrix Manipulations for Properties of Pell p-Numbers and their Generalizations

Abstract

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