Computing a specific coefficient in a product of polynomials
J, 21 19 18 bytes

{ ::0[:+//.@(*/)/&gt;


Try it online!

-1 byte thanks to FrownyFrog

J has a nice geometric idiom for multiplying polynomials, which we 
represent as lists of coefficients, with explicit zeroes where needed:

+//.@(*/)


Let's see how this works using the example:

1 3 5 +//.@(*/) 5 3 2


First it creates a multiplication table */:

 5  3  2
15  9  6
25 15 10


And @: then it computes the sums along each diagonal +//. using the Oblique /. adverb:



This works because moving down a diagonal is equivalent to decrementing the
power of one x while incrementing the power of another, meaning that the
numbers along a diagonal represent all the components of some a factor x^n
for some n.

The rest of the solution is just mechanics for the problem as stated:

{ ::0[: &lt;polynomial idiom&gt; /&gt;


Since we can have an arbitrary number of polynomials, we represent them as a list of boxes (each box containing one polynomial) and take that as the right arg.  The left arg is the index we want.

Now { ::0 ... is a dyadic hook asking for the index specified by the left arg, after transforming the right arg by everything in ....  And, if you can't find the index, return 0 ::0.

Finally [: &lt;polynomial idiom&gt; /&gt; says to first unbox &gt; the right arg lists, filling any missing coefficients with 0.  Then reduce / that list using the polynomial multiplication idiom.