On error sums formed by rational approximations with split denominators

In this paper we consider error sums of the form $$\sum_{m=0}^{\infty} \varepsilon_m\Big( \,b_m\alpha - \frac{a_m}{c_m}\,\Big) \,,$$ where $\alpha$ is a real number, $a_m$, $b_m$, $c_m$ are integers, and $\varepsilon_m=1$ or $\varepsilon_m ={(-1)}^m$. In particular, we investigate such sums for $$\alpha \in \big\{ \pi, e,e^{1/2},e^{1/3},\dots, \log (1+t), \zeta(2), \zeta(3) \big\}$$ and exhibit some connections between rational coefficients occurring in error sums for Ap\'ery's continued fraction for $\zeta(2)$ and well-known integer sequences. The concept of the paper generalizes the theory of ordinary error sums, which are given by $b_m=q_m$ and $a_m/c_m=p_m$ with the convergents $p_m/q_m$ from the continued fraction expansion of $\alpha$.