Title: The Generalized Fibonacci Sequences on an Integral Domain
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-21-00026; Volume 3 / Issue 2 / Year 2021, Pages 60-69
Document Type: Research Paper
aDepartment of Mathematics, Akdeniz University, 07058 Antalya, Turkey
Received: 1 April 2021, Accepted: 2 May 2021, Published: 23 May 2021.
Corresponding Author: Mustafa Alkan (Email address: firstname.lastname@example.org)
Full Text: PDF
In the literature, there are many sequences of both numbers and polynomials as Fibonacci numbers are generated. In this paper, we observe that it is not necessary to discriminate among them for statements and results of sequences of both numbers and polynomials.
Keywords: Integral domain, Generating Function, Generalized Fibonacci Polynomials, Generalized Lucas Polynomials, Generalized Humbert Polynomials, Binet FormulaReferences:
- Z. Akyuz and S. Halici, Some identities deriving from the nth power of a special matrix, Adv. in Dif. 2012 (23), 2012.
- Z. Akyuz and S. Halici, On some combinatorial identities involving the term of generalized Fibonacci and Lucas sequences, Hacettepe J. Math. 42 (4), 431–435, 2013.
- G. E. Bergum and V. E. Hoggatt, Sums and products for recurring sequences, The Fibonacci Quarterly 13 (2), 115–120, 1975.
- G. S Cheon, H. Kim and L. W. Shapiro, A generalization of Lucas polynomial sequence, Discrete Appl. Math. 157 (5), 920–927, 2009.
- K. A Dilcher, Generalization of Fibonacci polynomials and a representation of Gegenbauer polynomials of integer order, Fibonacci Quart. 25 (4), 300–303, 1987.
- S. Falcón, On the k-Lucas numbers, Int. J. Contemp Math. Sci. 6 (21-24), 1039–1050, 2011.
- S. Falcón and Á. Plaza, On the Fibonacci k-numbers, Chaos Solitons Fractals 32 (5), 1615–1624, 2007.
- S. Falcon and Á. Plaza, The k-Fibonacci sequence and the Pascal 2-triangle Chaos, Solitons & Fractals 33 (1), 38–49, 2007.
- S. Falcon and Á. Plaza, The k-Fibonacci hyperbolic functions, Chaos, Solitons & Fractals 38 (2), 409–420, 2008.
- S. Falcon and Á. Plaza, On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals 39, 1005–1019, 2009.
- H. Feng and Z. Zhang, Computational formulas for convoluted generalized Fibonacci and Lucas numbers, Fibonacci Quart. 41 (2), 144–151, 2003.
- A. F. Horadam, Basic properties of a certain generalized sequence of numbers, The Fibonacci Quarterly 3 (2), 161–176, 1965.
- A. F. Horadam, Tschebyscheff and other functions associated with the sequence Wn(a, b; p, q), The Fibonacci Quarterly 7 (1), 14–22, 1969.
- A. F. Horadam, A synthesis of certain polynomial sequences In: G. E. Bergum, A. N. Philippou, A. F. Horadam, editors, Applications of Fibonacci numbers, Vol. 6, Kluwer Academic Publishers, 215–229, Dordrecht, 1996.
- L. C. Hsu, On Stirling-type pairs and extended Gegenbauer-Humbert-Fibonacci polynomials, In: G. E. Bergum, A. N. Philippou, A. F. Horadam, editors. Applications of Fibonacci numbers, Vol. 5., Kluwer Academic Publishers, 367–377, Dordrecht, 1993.
- L. C. Hsu and P. J. S. Shiue, Cycle indicators and special functions, Ann. Comb. 5 (2), 179–196, 2001.
- N. Jacobson, Basic algebra. I., (2nd ed.), W. H. Freeman and Company, New York, 1985.
- M. Janjic, Hessenberg matrices and integer sequences, J. Integer Seq. 13 (7),10 pp, Article 10.7.8, 2010.
- M. Janjic, Determinants and recurrence sequences, J. Integer Seq. 15 (3), 21 pp., Article 12.3.5, 2012.
- T. Koshy, Fibonacci and Lucas Numbers with applications, Wiley-Interscience Publications, 2001.
- G. Lee and M. Asci, Some properties of the (p, q)-Fibonacci and (p, q)-Lucas polynomials, J. Appl. Math., 18 pp, Art. ID 264842, 2012.
- T. Mansour, A formula for the generating functions of powers of Horadam’s sequence, Australasian J. of Combinatorics 30, 207–212, 2004.
- A. Nalli and P. Haukkanen, On generalized Fibonacci and Lucas polynomials, Chaos Solitons Fractals 42 (5), 3179–3186, 2009.
- G. Ozdemir and Y. Simsek, Generating functions for two-variable polynomials related to a family of Fibonacci type polynomials and numbers, Filomat 30 (4), 969–975, 2016.
- G. Ozdemir, Y. Simsek and V. G. Milovanovic, Generating functions for special polynomials and numbers including Apostol-type and Humbert-type polynomials, Mediterr. J. Math. 14 (3), 1–17, 2017.
- J. L. Ramírez, Some properties of convolved k-Fibonacci numbers, ISRN Combin. Art. ID 759641, 2013.
- J. L. Ramírez , On convolved generalized Fibonacci and Lucas polynomials, Appl. Math Comput. 229, 208–213, 2014.
- N. Robbins, A new formula for Lucas numbers, Fibonaci Quaterly 29, 362–363, 1991.
- Z. H. Sun, On the properties of Newton-Euler pairs, J. Number Theory 114 (1), 88–123, 2005.
- H. Tian-Xiao and J. S. Peter, On sequences of numbers and polynomials defined by linear recurrence relations of order 2, Int. J. Math. Math Sci. 21, Art. ID 709386.356, 2009.
- M. E. Waddill and L. Sacks, Another generalized Fibonacci sequence, The Fibonacci Quarterly 5 (3), 209–222, 1967.
- J. Wang, Some new results for the (p, q)-Fibonacci and Lucas polynomials, Adv. Differ. Equ. 64, 15 pp, 2014.
- W. Weiping and W. Hui, Some results on convolved (p, q)-Fibonacci polynomials, Ing.Trans. and Special Function 26 (5), 340–356, 2015..
- W. Weiping and W. Hui, Generalized Humbert polynomials via generalized Fibonacci polynomials, Applied Mathematics and Computation 307, 204–216, 2017.
- Y. Yuan and W. Zhang, Some identities involving the Fibonacci polynomials, Fibonacci Quart. 40 (4), 314–318, 2002.
- W. Zhang, Some identities involving the Fibonacci numbers, Fibonacci Quart. 35 (3), 225–229, 1997.
- F. Z. Zhao and T. Wang, Some identities involving the powers of the generalized Fibonacci numbers, Fibonacci Quart. 41 (1), 7–12, 2003.