TY  - JOUR
T1  - Toward the Determination of Vietoris-like Polynomials
AU  - Gürses, Nurten
AU  - Çağlar Çay, Duygu
PY  - 2025
DA  - June
Y2  - 2025
DO  - 10.53570/jnt.1651994
JF  - Journal of New Theory
JO  - JNT
PB  - Naim ÇAĞMAN
WT  - DergiPark
SN  - 2149-1402
SP  - 10
EP  - 25
IS  - 51
LA  - en
AB  - This paper studies the relationship between polynomials and classical number sequences, focusing on their structural properties and mathematical significance. It explores a specific class of polynomials inspired by Vietoris' number sequences, referred to as Vietoris-like polynomials. The primary objective is to analyze their fundamental algebraic properties, recurrence relations, and special identities. The study employs algebraic methods to derive the recurrence relations and explicit formulas for these polynomials. Moreover, it establishes Catalan-like, Cassini-like, and d'Ocagne-like identities.
KW  - Polynomials
KW  - Vietoris
KW  - Vietoris-like polynomials
CR  - T. Koshy, Fibonacci and Lucas numbers with applications, Vol. 2, John Wiley & Sons, 2018.
CR  - M. Bicknell, A primer for the Fibonacci numbers: Part VII, The Fibonacci Quarterly 8 (4) (1970) 407-420.
CR  - M. Singh, O. Sikhwal, Y. Gupta, Generalized Fibonacci-Lucas polynomials, International Journal of Advanced Mathematical Sciences 2 (1) (2014) 81-87.
CR  - A. Boussayoud, M. Kerada, N. Harrouche, On the $k$-Lucas numbers and Lucas polynomials, Turkish Journal of Analysis and Number Theory 5 (4) (2017) 121-125.
CR  - Jr. V. E. Hoggatt, M. Bicknell, Roots of Fibonacci polynomials, The Fibonacci Quarterly 11 (3) (1973) 271-274.
CR  - P. Catarino, The $h(x)$-Fibonacci quaternion polynomials: Some combinatorial properties, Advances in Applied Clifford Algebras 26 (1) (2016) 71-79.
CR  - Jr. V. E. Hoggatt, M. Bicknell, Generalized Fibonacci polynomials, The Fibonacci Quarterly 11 (5) (1973) 457-465.
CR  - A. F. Horadam, Bro. J. M. Mahon, Pell and Pell-Lucas polynomials, The Fibonacci Quarterly 23 (1) (1985) 7-20.
CR  - A. F. Horadam, Jacobsthal representation polynomials, Significance 35 (2) (1997) 137-148.
CR  - L. Vietoris, Über das vorzeichen gewisser trigonometrischer summen (in German), Springer, 1958.
CR  - L. Vietoris, Eine verallgemeinerung der gleichung $(n+ 1)!= n!(n+ 1)$ und zugehörige vermutete ungleichungen, Monatshefte für Mathematik 97 (2) (1984) 157-160.
CR  - I. Caçao, M. I. Falcao, H. Malonek, Hypercomplex polynomials, Vietoris' rational numbers and a related integer numbers sequence, Complex Analysis and Operator Theory 11 (5) (2017) 1059-1076.
CR  - M. I. Falcao, H. R. Malonek, A note on a one-parameter family of non-symmetric number triangles, Opuscula Mathematica 32 (4) (2012) 661-673.
CR  - N. J. A. Sloane, The On-line Encyclopedia of Integer Sequences (1964), http://oeis.org/, Accessed 10 Jan 2025.
CR  - I. Caçao, M. I. Falcão, R. Helmuth, On generalized Vietoris' number sequences, Discrete Applied Mathematics 269 (2019) 77-85.
CR  - I. Caçao, M. I. Falcão, H. R. Malonek, On Vietoris' number sequence and combinatorial identities with quaternions, in: Vigo-Aguiar, J. (Ed.), 17th International Conference on Computational and Mathematical Methods in Science and Engineering, Almería, 2017, pp. 480-488.
CR  - P. Catarino, R. Almedia, On a quaternionic sequence with Vietoris' numbers, Filomat 35 (4) (2021) 1065-1086.
CR  - P. Catarino, R. Almeida, A note on Vietoris' number sequence, Mediterranean Journal of Mathematics 19 (1) (2022) 1-19.
CR  - S. Halıcı, Z. B. Gür, A note on weighted sums of Vietoris' sequence, Mathematica Montisnigri 61 (2024) 44-57.
CR  - I. Caçao, M. I. Falcão, H. R. Malonek, F. Miranda, G. Tomaz, On Appell-Vietoris polynomials, in: O. Gervasi, B. Murgante, C. Garau, D. C. Taniar, A. M. A. Rocha, M. N. Faginas Lago (Eds.), International Conference on Computational Science and Its Applications, Cham, 2024, pp. 302-316.
CR  - M. Özdemir, Introduction to hybrid numbers, Advances in Applied Clifford Algebras 28 (1) (2018) 11.
CR  - N. Gürses, G. Y. Saçlı, S. Yüce, On Vietoris' hybrid number sequence, Turkish Journal of Mathematics 48 (4) (2024) 658-672.
UR  - https://doi.org/10.53570/jnt.1651994
L1  - https://dergipark.org.tr/en/download/article-file/4664175
ER  -