Modular form quick question translation algebra

[binbagsss]

Please see attached.

I am trying to show that

$T_{p} f (\tau + 1) = T_{p} f (\tau )$
$f(\tau) \in M_k$ and so can be written as a expansion as $f(\tau)=\sum\limits^{\infty}_{0}a_{n}e^{2 \pi i n \tau }$
$f(\tau + 1) = f(\tau)$ since $e^{2\pi i n} = 1$
Similarly $f(p\tau + p) = f(p\tau)$ for the same reason since $np \in Z \geq 1$ so the extra exponential term is $1$ again.

But I DONT UNDERSTAND how it goes from $f(\frac{\tau + 1 + j}{p}) = f(\frac{\tau+j}{p})$ , since it is not guarenteed that $1/p$ is an integer, I mean it only is when $p=1$ so $e^{2 \pi n i (t+1+j)/p} = e^{ 2 \pi i n (t+j)/p}e^{2 \pi i n (1/p) }$ and $e^{2 \pi i n (1/p) }$ is equal to $1$ only when $p=1$. second equality of attached.

help please. thank you.

 

[fresh_42]

It is simply a substitution of $j \rightarrow i-1$ and calling the $i$ afterwards $j$ again. (A usual business when dealing with sums.)