Assuming $N$ is the number of securities in the index and $w_i$ is the weight of security $i$, one can rewrite $TR_{1,0}$, the total return of the index from time $0$ through $1$, as
$$
\begin{align*}
TR_{1,0}
&= \sum_{i=1}^Nw_i \frac {(p_{1,i}-p_{0,i}+c_{1,i})}{p_{0,i}} &&<\text{definition}> \\
&= \sum_{i=1}^Nw_i (p_{1,i}/p_{0,i}-p_{0,i}/p_{0,i}+c_{1,i}/p_{0,i}) &&<\text{algebra}> \\
&= \sum_{i=1}^Nw_i (p_{1,i}/p_{0,i}-1+c_{1,i}/p_{0,i}) &&<\text{algebra}> \\
&= \sum_{i=1}^Nw_i (pr_{1,i}+cr_{1,i}) &&<\text{substitute }pr_{1,i}=p_{1,i}/p_{0,i}-1, cr_{1,i}=c_{1,i}/p_{0,i}> \tag 1
\end{align*}
$$
where $p_{j,i}$ is the price of security $i$ at time $j$, $c_{j,i}$ is the income of security $i$ from time $j-1$ through time $j$, $pr_{j,i}$ is the price return of security $i$ from time $j-1$ through $j$, $cr_{j,i}$ is the income return of security $i$ from time $j-1$ through $j$.
Equation (1) can be generalized as
$$
TR_{j,j-1} = \sum_{i=1}^Nw_i (pr_{j,i}+cr_{j,i}) \tag 2
$$
where $TR_{j,j-1}$ is the total return of the index from time $j-1$ through time $j$.
The total return of the index from time 0 through time $K$ (the multi-period return) can be calculated by compounding single period returns. This can be expressed as
$$
\begin{align*}
TR_{K,0} &= \left( \prod\limits_{j=1}^{K} (TR_{j,j-1}+1) \right) - 1 &&<\text{definition}> \\
&= \left( \prod\limits_{j=1}^{K}
\left( \left( \sum_{i=1}^N w_i (pr_{j,i}+cr_{j,i}) \right) +1 \right)
\right) - 1. &&<\text{substitute equation (2)}> \tag 3
\end{align*}
$$
It is possible to generalize equations (2) and (3) further by assuming weights can change dynamically. Dynamic weights can be expressed as $w_{j-1,i}$ which would be the weight of security $i$ at time $j-1$. Then equation (3) can be rewritten as
$$
TR_{K,0} = \left( \prod\limits_{j=1}^{K}
\left( \left( \sum_{i=1}^N w_{j-1,i} (pr_{j,i}+cr_{j,i}) \right) +1 \right)
\right) - 1. \tag 4
$$
