On the Francois Polynomials and Hybrinomials

Year 2025, Volume: 13 Issue: 2, 117 - 124, 31.10.2025

Efruz Özlem Mersin

Abstract

This paper introduces the Francois polynomials and Francois hybrinomials, which are generalized forms of the Francois numbers and Francois hybrid numbers, respectively. We explore their recurrence relations, summation formulas, generating functions, and Binet-type formulas. Moreover, we derive the Catalan, Cassini, and d'Ocagne identities for these polynomials and hybrinomials.

Keywords

Binet formula , Catalan identity , d'Ocagne identity , Francois hybrinomials , Francois polynomials , Generating function

References

• [1] N. R. Ait-Amrane, H. Belbachir and E. Tan, On generalized Fibonacci and Lucas hybrid polynomials, Turkish Journal of Mathematics, 46(6) (2022), 2069-2077.
• [2] Y. Alp, On the Generalized Francois Numbers, Turkish Journal of Mathematics and Computer Science, 16(2) (2024), 346-353.
• [3] Y. Alp, Some Properties of the Generalized Leonardo Numbers, Journal of New Theory, 47 (2024), 52-60.
• [4] Y. Alp and E. G. Kocer, Hybrid Leonardo numbers, Chaos, Solitons & Fractals, 150 (2021), 111128.
• [5] M. Bicknell, An introduction to Fibonacci polynomials and their divisibility properties, Fibonacci Quarterly, 8(4) (1970), 407-420.
• [6] G. Bilgici, New generalizations of Fibonacci and Lucas sequences, Applied Mathematical Sciences, 8(29) (2014), 1429-1437.
• [7] D. Brod, A. Szynal-Liana and I. Włoch, On a new generalization of Pell hybrid numbers, Annales Mathematicae Silesianae, 38(2) (2024), 221-240.
• [8] E. C. Catalan, Note sur une ´equation aux diff´erences finies, Journal de Math´ematiques Pures et Appliqu´ees, 3 (1838), 508–516.
• [9] P. Catarino, On k-Pell hybrid numbers, Journal of Discrete Mathematical Sciences and Cryptography, 22(1) (2019), 83-89.
• [10] P. M. Catarino and A. Borges, On leonardo numbers, Acta Mathematica Universitatis Comenianae, 89(1) (2019), 75-86.
• [11] A. Dil and I. Mez˝o, A symmetric algorithm for hyperharmonic and Fibonacci numbers, Applied Mathematics and Computation, 206(2) (2008), 942-951.
• [12] O. Dis¸kaya, H. Menken and P. M. M. C. Catarino, On the hyperbolic Leonardo and hyperbolic Francois quaternions, Journal of New Theory, (42) (2023), 74-85.
• [13] M. Edson and O. Yayenie, A New Generalization of Fibonacci Sequence & Extended Binet’s Formula, Integers, 9(6) (2009), 639-654.
• [14] R. Fl´orez, D. Higuita and A. Ram´ırez, Catalan identities for generalized Fibonacci polynomials, Journal of Integer Sequences, 27 (2024), 24.1.4.
• [15] V. E. Hoggatt Jr. and M. Bicknell, Roots of Fibonacci polynomials, Fibonacci Quarterly, 11(3) (1973), 271-274.
• [16] C. Kızılates¸, A new generalization of Fibonacci hybrid and Lucas hybrid numbers, Chaos, Solitons & Fractals, 130 (2020), 109449.
• [17] C. Kızılates¸, A note on Horadam hybrinomials, Fundamental Journal of Mathematics and Applications, 5(1) (2022), 1-9.
• [18] C. Kızılates¸, W. S. Du and N. Terzio˘glu, On Higher-Order Generalized Fibonacci Hybrinomials: New Properties, Recurrence Relations and Matrix Representations, Mathematics, 12(8) (2024), 1156.
• [19] T. Koshy, Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics, A Wiley-Interscience Series of Texts, Monographs and Tracts, New York: Wiley 2001.
• [20] T. Koshy, Polynomial extensions of the Lucas and Ginsburg identities, Fibonacci Quarterly, 52(2) (2014), 141-147.
• [21] K. Kuhapatanakul and J. Chobsorn, On the generalized Leonardo numbers, Integers: Electronic Journal of Combinatorial Number Theory, 22 (2022).
• [22] F. Kürüz, A. Dagdeviren and P. Catarino, On Leonardo Pisano hybrinomials, Mathematics, 9(22) (2021), 2923.
• [23] M. Liana, A. Szynal-Liana and I. Włoch, On Pell hybrinomials, Miskolc Mathematical Notes, 20(2) (2019), 1051-1062.
• [24] E. O. Mersin, Hybrinomials Related to Hyper-Fibonacci and Hyper-Lucas Numbers, Journal of Engineering Technology and Applied Sciences, 8(1) (2023), 1-13.
• [25] E. O. Mersin, Hyperbolic Horadam hybrid functions, Notes on Number Theory and Discrete Mathematics, 29(2) (2023), 389-401.
• [26] E. O. Mersin and M. Bahs¸i, Hyper-Fibonacci and hyper-Lucas hybrinomials, Konuralp Journal of Mathematics, 10(2) (2022), 293-300.
• [27] E. O. Mersin and M. Bahs¸i, Hyper-Leonardo numbers, Conference Proceedings of Science and Technology, 5(1) (2022), 14-20.
• [28] E. O. Mersin and M. Bahs¸i, Hybrinomials related to hyper-Leonardo numbers, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(1) (2023), 240-246.
• [29] E. O. Mersin and M. Bahs¸i, Hyper-Leonardo hybrinomials, Eskisehir Technical University Science and Technology B-Theoretical Science, 11(1) (2023), 91-103.
• [30] M. Ozdemir, Introduction to hybrid numbers, Advances in Applied Clifford Algebras, 28 (2018), 1-32.
• [31] S. Petroudi, M. Pirouz and A. Ozkoc¸, The Narayana polynomial and Narayana hybrinomial sequences, Konuralp Journal of Mathematics, 9(1) (2021), 90-99.
• [32] J. Rooney, On the three types of complex number and planar transformations, Environment and Planning B: Planning and Design, 5(1) (1978), 89-99.
• [33] A. G. Shannon, A note on generalized Leonardo numbers, Notes on Number Theory and Discrete Mathematics, 25(3) (2019), 97-101.
• [34] A. G. Shannon and O. Deveci, A note on generalized and extended Leonardo sequences, Differences, 10(1) (2022), 1-12.
• [35] Y. Soykan, Generalized Leonardo numbers, Journal of Progressive Research in Mathematics, 18(4) (2021), 58-84.
• [36] A. Szynal-Liana, The Horadam hybrid numbers, Discussiones Mathematicae General Algebra and Applications, 38(1) (2018), 91-98.
• [37] A. Szynal-Liana and I. Włoch, Introduction to Fibonacci and Lucas hybrinomials, Complex Variables and Elliptic Equations, 65(10) (2020), 1736-1747.
• [38] A. Szynal-Liana and I. Włoch, On Pell and Pell-Lucas Hybrid Numbers, Commentationes Mathematicae, 58(1-2) (2018), 11-17.
• [39] A. Szynal-Liana and I. Włoch, Generalized Fibonacci-Pell hybrinomials, Online Journal of Analytic Combinatorics, 15 (2020), 1-12.
• [40] E. Tan and N. R. Ait-Amrane, On a new generalization of Fibonacci hybrid numbers, Indian Journal of Pure and Applied Mathematics, 54(2) (2023), 428-438.
• [41] O. Yayenie, A note on generalized Fibonacci sequences, Applied Mathematics and Computation, 217(12) (2011), 5603-5611.