  Let $\beta>1$ be fixed. We consider the $(\frak{b, d})$ numeration system,
where the base ${\frak b}=(b_k)_{k\geq 0}$ is a sequence of positive real
numbers satisfying $\lim_{k\rightarrow \infty}b_{k+1}/b_k=\beta$, and the set
of digits ${\frak d}\ni 0$ is a finite set of nonnegative real numbers with at
least two elements. Let $r_{\frak{b, d}}(\lambda)$ denote the number of
representations of a given $\lambda\in\mathbb{R}$ by sums $\sum_{k\ge
0}\delta_kb_k$ with $\delta_k$ in ${\frak d}$. We establish upper bounds and
asymptotic formulas for $r_{\frak{b,d}}(\lambda)$ and its arbitrary moments,
respectively. We prove that the associated zeta function $\zeta_{\frak{b,
d}}(s):=\sum_{\lambda>0}r_{\frak{b, d}}(\lambda)\lambda^{-s}$ can be
meromorphically continued to the entire complex plane when $b_k=\beta^{k}$, and
to the half-plane $\Re(s)>\log_\beta |\frak{d}|-\gamma$ when
$b_k=\beta^{k}+O(\beta^{(1-\gamma)k})$, with any fixed $\gamma\in(0,1]$,
respectively. We also determine the possible poles, compute the residues at the
poles, and locate the trivial zeros of $\zeta_{\frak{b, d}}(s)$ in the regions
where it can be extended. As an application, we answer some problems posed by
Chow and Slattery on partitions into distinct terms of certain integer
sequences.
