No. 1047

Title:

Bivariate Fibonacci polynomials of order $k$ with statistical applications

Author(s):

Kiyoshi, Inoue (Faculty of Economics, Seikei University);
Sigeo, Aki (Faculty of Engineering, Kansai University)

Key words:

Fibonacci polynomials, Lucas polynomials, success runs, waiting time, distributions of order $k$, probability generating function, continued fraction

Abstract:

    In the present article, we investigate the properties of bivariate Fibonacci polynomials of order $k$ in terms of the generating functions. For $k$ and $\ell $ ($1 \leq \ell \leq k-1 $), the relationship between the bivariate Fibonacci polynomials of order $k$ and the bivariate Fibonacci polynomials of order $\ell $ is elucidated. Lucas polynomials of order $k$ are considered. We also reveal the relationship between Lucas polynomials of order $k$ and Lucas polynomials of order $\ell $. The present work extends several properties of Fibonacci and Lucas polynomials of order $k$, which will lead us a new type of geneses of these polynomials. We point out that Fibonacci and Lucas polynomials of order $k$ are closely related to distributions of order $k$ and show that the distributions possess properties analogous to the bivariate Fibonacci and Lucas polynomials of order $k$.