On several generalizations of the Fibonacci numbers with application to population problems

Abstract

The present paper introduces an hierarchy (Figure \ref{fig:1}) of extensions of Fibonacci numbers (FN), that are motivated by generalizations of the associated population problem. The original Fibonacci numbers $f_n$ are the population at generation $n$, assuming that the progenitors have (i) unit offspring only, at (ii) the next generation, and (iii) live forever. Making explicit the three assumptions (i, ii, iii), points immediately to the three generalizations: the progenitors have (iii) a lifetime of $q$ generations, and (i) produce $k$ offspring, during (ii) each of the first $p\leq q$ generations. This leads to the triply extended Fibonacci numbers (TEFN) denoted by $g_n(k,p,q)$, of which the original Fibonacci numbers (FN) are the particular case $f_n=g_n(1,1,\infty)$. There are intermediate cases, namely three doubly extended Fibonacci numbers (DEFN) e.g. $c_n(k,q) \equiv g_n(k,p,\infty)$, and three singly extended Fibonacci numbers (EFN), e.g. $d_n(q) \equiv g_n(1,1,q)$. All these problems lead to finite difference equations, some of that have closed form solutions; the tables indicate several of the extended Fibonacci numbers which are relevant to population growth problems. Some of the populations problems go beyond the triply extended Fibonacci numbers, e.g. in the case of different birth rates at each generation, or percentage death rates at succeeding generations.