Special transforms of the generalized bivariate Fibonacci and Lucas polynomials

This paper deals with the Catalan, Hankel, binomial transforms of the generalized bivariate Fibonacci and Lucas polynomials. Also, some useful results such as generating functions, Binet formulas, summations of transforms defined by using recurrence relations of these special polynomials are presented. Furthermore, certain important relations among these transforms are deduced by using obtained new formulas. Finally, the Catalan, Cassini, Vajda and d'Ocagne formulas for these transforms are also derived.

Keywords:

binomial transform, bivariate Fibonacci polynomials, Catalan transform Hankel transform, Horadam polynomials,

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[1] H. Akkus, R. Uregen, E. Ozkan, A New Approach to k-Jacobsthal Lucas Sequences, SAUJS 25(4), 969-973, 2021.
[2] P. Barry, A Catalan transform and related transformations on integer sequences, J. Integer Seq. 8(4), Article 05.4.5, 24 pp, 2005.
[3] P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, J. Math. Comput. Sci. 8, 81-92, 2014.
[4] K. N. Boyadzhiev, Notes on the Binomial Transform, World Scientific, Singapore, 2018.
[5] K.W. Chen, Identities from the binomial transform, J. Number Theory 124, 142-150, 2007.
[6] A. Cvetkovic, P. Rajkovic, M. Ivkovic, Catalan numbers, the Hankel transform and Fibonacci numbers, J. Integer Seq. 5(1), 1-8, 2002.
[7] S. Falcon, Catalan Transform Of The k-Fibonacci Sequence, Commun. Korean Math. Soc. 28(4), 827-832, 2013.
[8] S. Falcon, A. Plaza, Binomial transforms of k-Fibonacci sequence, Int. J. Nonlinear Sci. Numer. Simul. 10(11-12), 1527-1538, 2009.
[9] R. Frontczak, Sums of the generalized bivariate Fibonacci and Lucas numbers with binomial coefficients, Int. J. Math. Anal. 12, 585-594, 2018.
[10] Y.K. Gupta, M. Singh, O. Sikhwal, Generalized Bivariate Fibonacci-Like Polynomials, MAYFEB Journal of Mathematics 1, 29-36, 2017.
[11] E.G. Kocer, S. Tuncez, Bivariate Fibonacci and Lucas Like Polynomials, Gazi Univ. J. Sci. 29(1), 109-113, 2016.
[12] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc. NY, 2001.
[13] J. W. Layman, The Hankel transform and some of its properties, J. Integer Seq. 4(1), Article 01.1.5, 11 pp, 2001.
[14] J. K. Mandal, S. K. Ghosal and J. Zizka, A Fragile Watermarking Based On Binomial Transform In Color Images, Computer Science and Information Technology, 281-288, 2013.
[15] A. Nalli, P. Haukkanen, On Generalizing Fibonacci and Lucas Polynomials, Chaos, Solitons & Fractals 42, 3179-3186, 2009.
[16] E. Ozkan, M. Tastan, O. Gungor, Catalan Transform of The k-Lucas Numbers, Erzincan University Journal of Science and Technology, Special Issue I-13, 145-149, 2020.
[17] E. Ozkan, M. Uysal, B. Kuloglu, Catalan transform of the incomplete Jacobsthal numbers and incomplete generalized Jacobsthal polynomials, Asian-Eur. J. Math., 2250119, 2021.
[18] E. Ozkan, B. Kuloglu, J.F. Peters, k-Narayana sequence self-Similarity flip graph views of k-Narayana self-Similarity, Chaos, Solitons & Fractals 153(2), 111473, 2021.
[19] H. Prodinger, Some information about the binomial transform, Fibonacci Q. 32(5), 412-415, 1994.
[20] P.M. Rajkovi´c, M.D. Petkovi´c, P. Barry, The Hankel transform of the sum of consecutive generalized Catalan numbers, Integral Transforms Spec. Funct. 18(4), 285-296, 2007.
[21] Y. Soykan, Binomial Transform of the Generalized Third Order Pell Sequence, Communications in Mathematics and Applications 12(1), 71-94, 2021.
[22] Y. Soykan, Notes on Binomial Transform of the Generalized Narayana Sequence, Earthline J. Math. Sci. 7(1), 77-111, 2021.
[23] M. Tastan, E. Ozkan, Catalan transform of the k-Jacobsthal sequence, Electronic Journal of Mathematical Analysis and Applications 8(2), 70-74, 2020.
[24] M. Tastan, E. Ozkan, Catalan transform of the k-Pell, k-PellLucas and modified k-Pell sequence, Notes Number Theory Discrete Math. 27(1), 198207, 2021.
[25] N. Tuglu, E.G. Kocer, A. Stakhov, Bivariate Fibonacci Like p-Polynomials, Appl. Math. Comput. 217(24), 10239-10246, 2011.
[26] S. Tuncez, Generalized Bivariate Fibonacci and Lucas polynomials, Ms Thesis, Konya, Turkey, 2011.
[27] S. Uygun, The binomial transforms of the generalized (s, t)-Jacobsthal matrix sequence, Int. J. Adv. Appl. Math. and Mech. 6(3), 14-20, 2019.
[28] J. Wang, Some new results for the (p, q)−Fibonacci and Lucas polynomials, Adv. Differ. Equ. 2014(1), 1-15, (2014).
[29] A. Włoch, Some identities for the generalized Fibonacci numbers and the generalized Lucas numbers, Appl. Math. Comput. 219(10), 5564-5568, 2013.
[30] N. Yilmaz, Binomial transforms of the balancing and Lucas-balancing polynomials, Contrib. Discrete Math. 15(3), 133-144, 2020.
[31] N. Yilmaz, N. Taskara, Binomial transforms of the Padovan and Perrin matrix sequences, Abstr. Appl. Anal. 2013, 2013.