Konuralp Journal of Mathematics, 6 (2) (2018) 209-212 Konuralp Journal of Mathematics Journal Homepage: www.dergipark.gov.tr/konuralpjournalmath e-ISSN: 2147-625X On Some Properties of Incomplete Trivariate Generalized Tribonacci Polynomials S¨ umeyra Uc¸ar 1* and Nihal Yılmaz ¨ Ozg ¨ ur 2 1 Department of Mathematics, Faculty of Science and Arts, Balıkesir University, Balıkesir, Turkey 2 Department of Mathematics, Faculty of Science and Arts, Balıkesir University, Balıkesir, Turkey * Corresponding author E-mail: sumeyraucar@balikesir.edu.tr Abstract In this paper we define incomplete trivariate generalized Tribonacci polynomials and obtain some properties of them using tables and sum formulas. Especially, we obtain a recurrence relation of these new class of polynomials. Keywords: Trivariate generalized Fibonacci polynomials; incomplete trivariate generalized Fibonacci polynomials. 2010 Mathematics Subject Classification: Primary 11B39; 11B37; 11B83. 1. Introduction From [6], we know that Tribonacci numbers studied in 1963 by M. Feinberg when he was a 14-years-old. Tribonacci numbers T n are defined by T 0 = 0, T 1 = 1, T 2 = 1 and T n = T n-1 + T n-2 + T n-3 for any integer n > 2. In [4], Tribonacci polynomials are defined by t 0 (x)= 0, t 1 (x)= 1, t 2 (x)= x 2 and t n (x)= x 2 t n-1 (x)+ xt n-2 (x)+ t n-3 (x) for any integer n > 2. In [5], trivariate Fibonacci polynomials defined by the following recurrence relations H n (x, y, z)= xH n-1 (x, y, z)+ yH n-2 (x, y, z)+ zH n-3 (x, y, z) with the initial conditions H 0 (x, y, z)= 0, H 1 (x, y, z)= 1, H 2 (x, y, z)= x. Note that H n (1, 1, 1)= T n and H n (x 2 , x, 1)= t n (x). In [2], it was given incomplete k-Pell, k-Pell-Lucas and modified k-Pell numbers and their some properties. In [3], it was given a proof of the conjecture given in [7] about generating function of the incomplete Tribonacci numbers and given Tribonacci polynomial triangle. In [5], it was obtained some properties of the trivariate Fibonacci and Lucas polynomials such as Binet formulas, generating functions of them (see for more details [5]). In this study we define trivariate generalized Tribonacci polynomials and incomplete trivariate generalized Tribonacci polynomials, then we investigate some properties of these polynomials and construct interesting tables of them. 2. Incomplete Trivariate Generalized Tribonacci Polynomials At first, we give the definition of trivariate generalized Tribonacci polynomials. Definition 2.1. Let p(x, y, z), q(x, y, z) and r(x, y, z) be polynomials with 3 real variables and real coefficients. Trivariate generalized Tribonacci polynomials are defined by the following recurrence relation: M n ( p, q, r)= p(x, y, z)M n-1 ( p, q, r)+ q(x, y, z)M n-2 ( p, q, r)+ r(x, y, z)M n-3 ( p, q, r), n ≥ 3 (2.1) with the initial conditions M 0 ( p, q, r)= 0, M 1 ( p, q, r)= 1 and M 2 ( p, q, r)= p(x, y, z). Email addresses: , sumeyraucar@balikesir.edu.tr (S¨ umeyra Uc¸ar), nihal@balikesir.edu.tr (Nihal Yılmaz ¨ Ozg ¨ ur)