Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑nk=0 xkWmk+j
Archives of Current Research International,
Page 1138
DOI:
10.9734/acri/2021/v21i330235
Abstract
In this paper, closed forms of the summation formulas ∑nk=0 xkWmk+j for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, PellLucas, Jacobsthal, JacobsthalLucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we give some identities and recurrence properties of generalized Fibonacci sequence.
Keywords:
 Fibonacci numbers
 Lucas numbers
 Pell numbers
 Jacobsthal numbers
 sum formulas
 recurrence properties
How to Cite
Soykan, Y. (2021). Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑nk=0 xkWmk+j. Archives of Current Research International, 21(3), 1138. https://doi.org/10.9734/acri/2021/v21i330235
References
Horadam AF. Basic properties of a certain generalized sequence of numbers. Fibonacci Quarterly. 1965;3(3):161176.
Horadam AF. A generalized Fibonacci sequence. American Mathematical Monthly. 1961;68:455459.
Horadam AF. Special properties of the sequence wn(a, b; p, q). Fibonacci Quarterly. 1967;5(5):424434.
Soykan; ACRI, 21(3): 1138, 2021; Article no.ACRI.69283
Horadam AF. Generating functions for powers of a certain generalized sequence of numbers. Duke Math. J. 1965;32:437446.
Soykan Y. On generalized (r,s)numbers. International Journal of Advances in Applied Mathematics and Mechanics. 2020;8(1):1 14.
Sloane NJA. The online encyclopedia of integer sequences.
Available:http://oeis.org/
Akbulak M, Oteles¸ A. On the Sum of ¨ Pell and Jacobsthal Numbers by Matrix Method. Bull. Iranian Mathematical Society. 2014;40(4):10171025.
Aydın FT. On generalizations of the Jacobsthal sequence. Notes on Number Theory and Discrete Mathematics. 2018;24(1):120135.
Catarino P, Vasco P, Campos APA, Borges A. New families of Jacobsthal and JacobsthalLucas numbers. Algebra and Discrete Mathematics. 2015;20(1):4054.
Cerin Z. Formulae for sums of Jacobsthal– ˇ Lucas numbers. Int. Math. Forum. 2007;2(40):19691984.
Cerin Z. Sums of squares and products ˇ of Jacobsthal numbers. Journal of Integer Sequences. 2007;10: Article 07.2.5.
Dasdemir A. On the Jacobsthal numbers by matrix method. SDU Journal of Science. 2012;7(1):6976.
Dasdemir A. A study on the Jacobsthal and Jacobsthal–Lucas numbers by matrix method. DUFED Journal of Sciences. 2014;3(1):1318.
Gnanam A, Anitha B. Sums of squares Jacobsthal numbers. IOSR Journal of Mathematics. 2015;11(6):6264.
Horadam AF. Jacobsthal representation numbers. Fibonacci Quarterly. 1996;34:40 54.
Horadam AF. Jacobsthal and pell curves. Fibonacci Quarterly. 1988;26:7783.
Kocer GE. Circulant, Negacyclic and Semicirculant Matrices with the Modified Pell, Jacobsthal and JacobsthalLucas Numbers. Hacettepe Journal of Mathematics and Statistics. 2007;36(2):133142.
Koken F, Bozkurt D. On the Jacobsthal ¨ numbers by matrix methods. Int. J. Contemp Math. Sciences. 2008;3(13):605614.
Mazorchuk V. New families of Jacobsthal and JacobsthalLucas numbers. Algebra and Discrete Mathematics. 2015;20(1):40 54.
Uygun, S¸ . Some Sum Formulas of (s, t) Jacobsthal and (s, t)Jacobsthal Lucas Matrix Sequences, Applied Mathematics. 2016;7:6169.
Uygun S. The binomial transforms of the generalized (s,t)Jacobsthal matrix sequence. Int. J. Adv. Appl. Math. and Mech. 2019;6(3):1420.
Bicknell N. A primer on the Pell sequence and related sequence. Fibonacci Quarterly. 1975;13(4):345349.
Dasdemir A. On the Pell, PellLucas and Modified Pell Numbers By Matrix Method, Applied Mathematical Sciences. 2011;5(64):31733181.
Ercolano J. Matrix generator of Pell sequence, Fibonacci Quarterly. 1979;17(1):7177.
Gokbas H, K ¨ ose H. Some sum formulas ¨ for products of Pell and PellLucas numbers. Int. J. Adv. Appl. Math. and Mech. 2017;4(4):14.
Horadam AF. Pell Identities. Fibonacci Quarterly. 1971;9(3):245263.
Soykan; ACRI, 21(3): 1138, 2021; Article no.ACRI.69283
Kilic¸ E, Tas¸c¸i D. The Linear Algebra of The Pell Matrix, Bolet´ın de la Sociedad Matematica Mexicana. 2005;3(11). ´
Koshy T. Pell and pellLucas numbers with applications. Springer, New York; 2014.
Melham R. Sums involving Fibonacci and Pell numbers, Portugaliae Mathematica. 1999;56(3):309317.
Kilic¸ E, Tas¸c¸i D. The Generalized Binet Formula, Representation and Sums of the Generalized Orderk Pell Numbers, Taiwanese Journal of Mathematics. 2006;10(6):16611670.
Kilic¸ E, Stanica P. A matrix approach for general higher order linear Recurrences. Bulletin of the Malaysian Mathematical Sciences Society. 2011;(2)34(1):5167.
Soykan Y. On Generalized ThirdOrder Pell Numbers. Asian Journal of Advanced Research and Reports. 2019;6(1):118.
Soykan Y. A study of generalized fourth order Pell sequences. Journal of Scientific Research and Reports. 2019;25(12):118.
Soykan Y. Properties of generalized fifth order Pell numbers. Asian Research Journal of Mathematics. 2019;15(3):118.
Soykan Y. On generalized sixthorder Pell sequence. Journal of Scientific Perspectives. 2020;4(1):4970.
Horadam AF. Basic properties of a certain generalized sequence of numbers. Fibonacci Quarterly. 1965;3(3):161176.
Horadam AF. A generalized Fibonacci sequence. American Mathematical Monthly. 1961;68:455459.
Horadam AF. Special properties of the sequence wn(a, b; p, q). Fibonacci Quarterly. 1967;5(5):424434.
Soykan; ACRI, 21(3): 1138, 2021; Article no.ACRI.69283
Horadam AF. Generating functions for powers of a certain generalized sequence of numbers. Duke Math. J. 1965;32:437446.
Soykan Y. On generalized (r,s)numbers. International Journal of Advances in Applied Mathematics and Mechanics. 2020;8(1):1 14.
Sloane NJA. The online encyclopedia of integer sequences.
Available:http://oeis.org/
Akbulak M, Oteles¸ A. On the Sum of ¨ Pell and Jacobsthal Numbers by Matrix Method. Bull. Iranian Mathematical Society. 2014;40(4):10171025.
Aydın FT. On generalizations of the Jacobsthal sequence. Notes on Number Theory and Discrete Mathematics. 2018;24(1):120135.
Catarino P, Vasco P, Campos APA, Borges A. New families of Jacobsthal and JacobsthalLucas numbers. Algebra and Discrete Mathematics. 2015;20(1):4054.
Cerin Z. Formulae for sums of Jacobsthal– ˇ Lucas numbers. Int. Math. Forum. 2007;2(40):19691984.
Cerin Z. Sums of squares and products ˇ of Jacobsthal numbers. Journal of Integer Sequences. 2007;10: Article 07.2.5.
Dasdemir A. On the Jacobsthal numbers by matrix method. SDU Journal of Science. 2012;7(1):6976.
Dasdemir A. A study on the Jacobsthal and Jacobsthal–Lucas numbers by matrix method. DUFED Journal of Sciences. 2014;3(1):1318.
Gnanam A, Anitha B. Sums of squares Jacobsthal numbers. IOSR Journal of Mathematics. 2015;11(6):6264.
Horadam AF. Jacobsthal representation numbers. Fibonacci Quarterly. 1996;34:40 54.
Horadam AF. Jacobsthal and pell curves. Fibonacci Quarterly. 1988;26:7783.
Kocer GE. Circulant, Negacyclic and Semicirculant Matrices with the Modified Pell, Jacobsthal and JacobsthalLucas Numbers. Hacettepe Journal of Mathematics and Statistics. 2007;36(2):133142.
Koken F, Bozkurt D. On the Jacobsthal ¨ numbers by matrix methods. Int. J. Contemp Math. Sciences. 2008;3(13):605614.
Mazorchuk V. New families of Jacobsthal and JacobsthalLucas numbers. Algebra and Discrete Mathematics. 2015;20(1):40 54.
Uygun, S¸ . Some Sum Formulas of (s, t) Jacobsthal and (s, t)Jacobsthal Lucas Matrix Sequences, Applied Mathematics. 2016;7:6169.
Uygun S. The binomial transforms of the generalized (s,t)Jacobsthal matrix sequence. Int. J. Adv. Appl. Math. and Mech. 2019;6(3):1420.
Bicknell N. A primer on the Pell sequence and related sequence. Fibonacci Quarterly. 1975;13(4):345349.
Dasdemir A. On the Pell, PellLucas and Modified Pell Numbers By Matrix Method, Applied Mathematical Sciences. 2011;5(64):31733181.
Ercolano J. Matrix generator of Pell sequence, Fibonacci Quarterly. 1979;17(1):7177.
Gokbas H, K ¨ ose H. Some sum formulas ¨ for products of Pell and PellLucas numbers. Int. J. Adv. Appl. Math. and Mech. 2017;4(4):14.
Horadam AF. Pell Identities. Fibonacci Quarterly. 1971;9(3):245263.
Soykan; ACRI, 21(3): 1138, 2021; Article no.ACRI.69283
Kilic¸ E, Tas¸c¸i D. The Linear Algebra of The Pell Matrix, Bolet´ın de la Sociedad Matematica Mexicana. 2005;3(11). ´
Koshy T. Pell and pellLucas numbers with applications. Springer, New York; 2014.
Melham R. Sums involving Fibonacci and Pell numbers, Portugaliae Mathematica. 1999;56(3):309317.
Kilic¸ E, Tas¸c¸i D. The Generalized Binet Formula, Representation and Sums of the Generalized Orderk Pell Numbers, Taiwanese Journal of Mathematics. 2006;10(6):16611670.
Kilic¸ E, Stanica P. A matrix approach for general higher order linear Recurrences. Bulletin of the Malaysian Mathematical Sciences Society. 2011;(2)34(1):5167.
Soykan Y. On Generalized ThirdOrder Pell Numbers. Asian Journal of Advanced Research and Reports. 2019;6(1):118.
Soykan Y. A study of generalized fourth order Pell sequences. Journal of Scientific Research and Reports. 2019;25(12):118.
Soykan Y. Properties of generalized fifth order Pell numbers. Asian Research Journal of Mathematics. 2019;15(3):118.
Soykan Y. On generalized sixthorder Pell sequence. Journal of Scientific Perspectives. 2020;4(1):4970.
Horadam AF. Basic properties of a certain generalized sequence of numbers. Fibonacci Quarterly. 1965;3(3):161176.

Abstract View: 78 times
PDF Download: 64 times
Download Statistics
Downloads
Download data is not yet available.