What is the suitable mathematical form of the multipath channel impulse response?
Let's say we have a function that we transmit which is:
$$x(t)$$
Transmitting the function is kinda easy part. We assume there is just the function we are trying to send.

However things are a little bit ugly in the air. Let's look at the receiver part.

What we generally consider first is the AWGN. Which is:
$$r(t) = y(t) + n(t)$$

$r(t)$ is our received function. $y(t)$ is our function that reaches our hand and we assume we don't know what it is. In this case, for now, please consider $y(t) = x(t)$. Lastly we have gaussian distributed noise which is: $n(t)$.

Well, we have some fading types should be added. Let's combine it together and call them Channel Response. Channel responses should vary over time but for this situation I will assume it is constant. 

Let's call this $h_k$. 

The subscript of $h$ basically means we have different channel response for each channels. And hence we have:
$$h*y(t) + n(t)$$

But, hey! I don't see that at the receiver! There is some odd peaks at the signal! What is that thing?

Sadly, this is multipath and, at the receiver, you are staring at this signal:

$$(\sum\limits_{k=1}^K = h_k*y(t)) + n(t)$$

Aaaand no! They are not coming at the same time. Let's add a delay $\tau_k$

$$(\sum\limits_{k=1}^K = h_k*y(t - \tau_k)) + n(t)$$

$K$ is the number of multipath. Please notice that as the multipath occurs, each path has it's own channel. As you can see, there are different channel responses and delay times for each.

I hope it will be useful.

As always, have a nice day.