I feel very strange asking this but here is a problem I have from a textbook of mine. Simply put, I do not understand how (a_(t-1)*a_(t-2)…a_1) became the pi-product of a_i/a_o.
As a result, I dont understand how he got to this definition for a difference equation
On
The book seems to be wrong, unless there's some hidden assumption not shown in your picture. For $t = 4$, the first line gives
$$ y_4 = (a_3a_2a_1a_0)y_0 + (a_3a_2a_1)b_0 + (a_3a_2)b_1 + (a_3)b_2 + b_3 $$
While the second line gives
$$\begin{align*} y_4 &= (a_3a_2a_1a_0)y_0 + b_0\left(\frac{a_3a_2a_1a_0}{a_0^4}\right) + b_1\left(\frac{a_3a_2a_1}{a_1^3}\right) + b_2\left(\frac{a_3a_2}{a_2^2}\right) + b_3\left(\frac{a_3}{a_3}\right) \\ &= (a_3a_2a_1a_0)y_0 + \frac{a_3a_2a_1}{a_0^3}\cdot b_0 + \frac{a_3a_2}{a_1^2}\cdot b_1 + \frac{a_3}{a_2}\cdot b_2 + b_3 \end{align*}$$
It is wrong: $$\prod_{i=0}^{t-1} \frac{a_i}{a_0} \not= \frac{1}{a_0} \prod_{i=0}^{t-1} a_i = \frac{1}{a_0} a_0 \cdots a_{t-1} = a_1 \cdots a_{t-1}$$ Correct notation is: $$\frac{\prod_{i=0}^{t-1} a_i}{a_0} = a_1 \cdots a_{t-1}$$
Note that it is corrected in the fourth edition of the book.
More generally: $$\prod_{i=0}^{t-1} k\cdot a_i = k^t \prod_{i=0}^{t-1} a_i$$ Contrast with sums: $$\sum_{i=0}^{t-1} k\cdot a_i = k \sum_{i=0}^{t-1} a_i$$