Applying Fast Fourier Transform with a Specific Product Function

Question

Problem: Provided 2Npower polynomials like and . Suppose that 2 coefficients are the same for 2 polynomials.a0+a1x+a2x2..+aNxNb0+b1x+b2x2..+bNxN

Find the product of polynomials, the catch is that the product operator (*) between the coefficients is determined differently:

f(i,j) := ai * bj := 1/abs(ai-bj)

This means that we find the following polynomial:

c0+c1x+c2x2..+c2Nx2N,

Where

ck := Σi=0..N f(i,k-i)

I know that using FFT 2 polynomials can be multiplied in O(n lg n)time with the normal product operator between the coefficients, but have no idea about this problem.