APPLICATION OF MARKOV CHAINS PROPERTIES TO r-GENERALIZED FIBONACCI SEQUENCES

Abstract

where V^..^Vr_x are specified by the initial conditions. A first connection between Markov chains and sequence (1), whose coefficients at (0 < / < r -1) are nonnegative, is considered in [6]. And we established that the limit of the ratio Vn I q exists if and only if CGD{/ +1; at > 0} = 1, where CGD means the common great divisor and q is the unique positive root of the characteristic polynomial P(x) = x -a0x~ ar_2x-ar_l (cf. [6] and [7]). Our purpose in this paper is to give a second connection between Markov chains and sequence (1) when the at are nonnegative. This allows us to express the general term Vn (n>r) in a combinatoric form. Note that the combinatoric form of Vn has been studied by various methods and techniques (cf. [2], [4], [5], [8], [9], and [10], for example). However, our method is different from those above, and it allows us to study the asymptotic behavior of the ratio Vn/q, from which we derive a new approximation of the number q. This paper is organized as follows. In Section 2 we study the connection between Markov chains and sequence (1) when the coefficients a, are nonnegative and a0 + ••• +ar_x 1, and we establish the combinatoric form of Vn for n > r. In Section 3 we are interested in the asymptotic behavior of Vn when the coefficients cij are arbitrary nonnegative real numbers.

Cite this paper

@inproceedings{Mouline1999APPLICATIONOM, title={APPLICATION OF MARKOV CHAINS PROPERTIES TO r-GENERALIZED FIBONACCI SEQUENCES}, author={Mehdi Mouline and M. Rachidi}, year={1999} }