What does $\sum_{k=a}^b y_k$ mean for difference equations?

Question

In the book of Difference Equations by Peterson, at page 25-26, it is given that

However, in the previous pages the indefinite sum of y(t) is denoted as $\sum y(t)$, and now they put indexed to sum symbol, and I'm confused about what exactly $\sum_{k=a}^b y_k$ means. is it the sum of the functions $y_k$s, or the indefinite sum of their sum, or like $\sum \sum ... \sum y_k$ ?, but then why the $indices$ ?

I'm looking for clarification about the notation.

Answer 1

I first thought there was a mistake in the notation, but apparently is not the case;

$$\Delta_n(\sum_{k=m}^{n-1} y_k) = \Delta_n( y_{n-1} + y_{n-2} +... y_m) = y_n,$$ where $m$ is a fixed number, so that the lower bound of the sum does not change with the difference operator.

Answer 2

There is an assumption, that the domain of $y(t)$ is $\mathbb{N}$. Therefore the function $y(t)$ could be represented by the sequence $\{y_n\}$, where $y_k=y(k)$.

$y_k$s are not indexed functions, but indexed values of the function $y$