Horadam Polynomials

U ovom radu upoznat ćemo se s polinomima zadanim rekurzivnim formulama. Navest ćemo nekoliko vrsta takvih polinoma te objasniti Binetovu formulu vezanu uz Fibonaccijeve polinome. Detaljnije ćemo obraditi dvije vrste polinoma, Pellove i Pell-Lucasove. Opisat ćemo njihova svojstva i grafički prikazati neke od polinoma. Osim Pell-Lucasovih, obradit ćemo i općenitiju vrstu, k-Pell-Lucasove polinome. Proučit ćemo pripadne matrične zapise te konvoluciju polinoma. Konačno, dokazat ćemo jednakosti koje povezuju Pellove i PellLucasove polinome. Navest ćemo jednakosti koje vrijede izmedu Fibonaccijevih, Pellovih i Pell-Lucasovih brojeva te opisati na koji način se oni mogu primijeniti u kombinatorici.In this paper we will introduce type of polynomials defined by the recursive formulas. We will mention some of these polynomials and explain Binet formula. Furthermore, we will present two of the polynomials, Pell and Pell-Lucas, in detail. Some properties and graphical representation will also be shown. Besides that, k-Pell-Lucas polynomials will be mentioned. We will see some matrix techniques for these polynomials, as well as convolution. Finally, equalities that link Pell and Pell-Lucas polynomials will be proven. After that, we will mention the numbers which are obtained through certain polynomials. Some equalities will be used to prove connection between these numbers and trigonometric functions. We will end this paper with application of these numbers in Combinatorics