On waiting time distribution of runs in a Fibonacci and Lucassequences

Abstract

There is an increasingly needed to development a new mathematical apparatus concerned with the description, prediction, andunderstanding of natural phenomena in a precise manner. The purpose of this study is to extend the mathematical framework ofFibonacci and Lucas sequences for underpinned and establishing a modern mathematical formulas. For this purpose, we deriveanalytical formulas of order k-Fibonacci and order k-Lucas sequences simultaneously based on Bernoulli sequence. Furthermore,exploiting a relationship with the k th order Fibonacci and Lucas sequence, we study the probability distribution function (pdf) of thewaiting time (W (k)).