Asymptotic behaviour of the partition functionстатья
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Дата последнего поиска статьи во внешних источниках: 18 июля 2013 г.
Аннотация: Let $m,d$ be given positive integers such that $2 \leq m
\leq d$. For an $n \ge 0$ the partition function $b_{m,d}(n)$ is
the cardinality of the set $$ \{(a_0,a_1,\ldots)| n = \sum_k
a_km^k,\ a_k \in \{0,\ldots,d-1\}, k \geq 0\}. $$ In this paper we
analyze the asymptotic behavior of $b_{m,d}(n)$ as $n \to \infty$.
Using linear algebraic tools we prove the existence of positive
constants $C_1,C_2$ depending on $m$ and $d$ such that $$
C_1n^{\lambda_1}\leq b_{m,d}(n)\leq C_2n^{\lambda_2},\quad
n\in\Bbb N, $$ where
$\lambda_1=\mathop{\underline{\lim}}\limits_{n\rightarrow\infty}\,\,\frac
{\log b(n)}{\log
n},\quad\lambda_2=\mathop{\overline{\lim}}\limits_
{n\rightarrow\infty}\,\,\frac{\log b(n)}{\log n}$ are growth
exponents of the partition function. We find explicitly these
exponents for some pairs $(m,d)$, for the other pairs we express
them by the joint spectral radius of suitable linear operators.
Also we obtain estimations for the constants $C_1$ and $C_2$.