Regarding proving linear independence of product $G_k-12r$×$\Delta^r$

Question

I am self studying Apostol modular functions and Dirichlet series in number theory and I have a doubt in text of Chapter -6

Apostol mentions that it is easy to prove that products $G_{k-12r}$×$\Delta^r$ are Linearly Independent using $\Delta(i \infty) $ =0 but $G_{2r}(i \infty ) \neq 0 $ .

But if I assume $c_r $ × $G_{k-12r}$×$\Delta^r$ =0 and put $\tau$ = i$\infty $ then I will get both RHS and LHS =0 .

Edit 1 -> The statement which I don't know how to prove is related to proving this theorem.

So, can someone please tell how to prove their linear independence.

Answer 1

The question is about Theorem 6.3 in Section 6.4 on page 117. The proof ends on page 118 as follows

To show there is at most one such representation we need only verify that the products $G_{k-12r}\Delta^r$ are linearly independent. This follows easily from the fact that $\Delta(i\infty)=0$ but $G_{2r}(i\infty)\ne0$. details are left as an exercise for the reader.

As a comment by Lord Shark indicates, the explanation comes from the $q$ expansion of the products. That is, if $\tau=i\infty$ then $q=0$. The statement $G_{2r} i\infty)\ne0$ comes from equation $(48)$ on page 69 where $G_{2k} = 2\zeta(2k)$ plus higher order terms in the $q$ expansion. The other fact comes from equation $(8)$ on page 51 where $\Delta(\tau) = (2\pi)^{12}$ times a $q$ series which begins with $q$. These two facts give us that $G_{k-12}\Delta^r$ has a leading term of $q^r$ times a constant. Any sequence of functions with this property will be linearly independent. This is analogous to the statement that any sequence of polynomials of different degrees will be linearly independent and the proof is essentially the same.