Increasing Self-Described Sequences

Abstract.

We proceed with our study of increasing self-described sequences F, beginning with 1 and defined by a functional equation $|F^{ - 1} (m)| = \prod\nolimits_{0 \leq i \leq k} {F_i (m)^{a_i } (1 + o(1))} \;(a_i \geq 0\;{\text{and}}\;F_i \;{\text{denoting}}\;F \circ \cdots \circ F).$ In [1] we exhibited the simple solution f′ (t)=Ct^β, for some β ∈(0,1), of the associated functional-differential equation $f'(t) = \prod\nolimits_{0 \leq i \leq k} {f_i (t)^{ - a_i } } ,$ and we proved that provided β<2/(2+d(Γ)), where $d(\Gamma ): = a_1 + \cdots + a_k ,$ we have the asymtotic equivalence F(m)~ Cm^β.

In the present paper we show that this last result is optimal, in the sense that the self-described sequence defined by |F⁻¹(m)|=F(m)², that is

$$ 1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,7, \ldots , $$

for which the boundary case β=2/(2+d(Γ))(=1/2) holds, does not satisfy F(m) ~ Cm^β. We also show that the m-th term F(m) of a sequence F for which the boundary case holds is nevertheless of asymptotic order m^β.

Then we investigate the behaviour of self-described sequences F when β lies beyond the boundary case. In [1] we established the estimates $m^{\beta - \varepsilon } \ll F(m) \ll m^{\beta + \varepsilon } (*)$ when β is the unique fixed point of a certain associated function. We were only able to prove in general that the latter holds when β does not lie beyond the boundary case, however. In the present paper we prove that whenever $\beta \leq {1 \mathord{\left/ {\vphantom {1 {\sqrt {1 + d(\Gamma )} }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 + d(\Gamma )} }},$ β is the unique fixed point of this function, and in addition we obtain estimates more precise than (*). This applies for instance to the sequence defined by $|F^{ - 1} (m)| = (F \circ F)(m),$ that is

$$ 1,2,2,3,3,4,4,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,10,11, \ldots .. $$