I'm reading a paper, specifically "Forecasting correlated time series with exponential smoothing models" by Corberan-Vallet et al, and I came across this little beauty: A lower triangular $n\times n$ Matrix $L$ whose entries below the main diagonal are given by $$ L_{ij}=\alpha+\alpha\beta(i-j)+\gamma(i=j\text{ mod } s) $$ such that $j < i$
I don't understand the $i = j \text{ mod } s$ part. What does this mean? $s$ in this context is a known fixed integer, and $\alpha$, $\beta$, and $\gamma$ are real constants such that $0 \leq \alpha, \beta, \gamma \leq 1$. The confusing part of this for me is that i, j, and s are known. Does this work similarly to a boolean where $\gamma$ will only be added so long as the equality holds?
Thanks in advance!
(Typing an answer because I can't comment yet.) That equality should be a congruence. My best bet is that this is a typo,
$$ L_{ij}=\alpha+\alpha\beta(i-j)+\gamma \ \ (i \equiv j\text{ mod } s) $$
It shows that $i$ is always congruent to $j \mod s$. $ j < i$ would follow from $ s < i$ if that was ever implied before. Without any more context, I don't really see why it would be written as shown.
(Edit) I just read that portion of the paper, and it says 'for' not 'such that', so my original thoughts don't line up with that information. It could then be an implicit way to say for $s < i$ or something along that line.
(Last edit) Given your information, I'd prefer this:
$$ L_{ij}=\alpha+\alpha\beta(i-j)+\gamma f(i) $$
Where,
$$ f(i) = \begin{cases} 1 &\text{ if } i \equiv j \mod s\\ 0 &\text{ if } i \not\equiv j \mod s\end{cases} $$