I am looking out to simplify the following double summation in $\mathbb{F}_q[x_1,x_2]$, where $p$ is a prime and $q=p^k$ for some positive integer $k$ and a positive integer $r$ such that $0 \leq r \leq q-2$:
$\sum\limits_{i=0}^{q-2} \sum\limits_{j=0}^r {(q-i)(q-1) \choose j}x_1^{(q-i+j)(q-1)} {(i+2)(q-1) \choose r-j} x_2^{(i+2+r-j)(q-1)}$
(Assume that the sum is taken over only those terms where the binomial coefficients make sense) This sum arose during my study of invariant theory, on the application of Steenrod operations to certain polynomials in $\mathbb{F}_q[x_1,x_2]$ (you can comment if you want more details). I expect the sum to be a non-zero scalar multiple of the following sum:
$\sum\limits_{i=0}^{q-r-2}x_1^{(q-i)(q-1)}x_2^{(i+r+2)(q-1)}$
I tried changing the order of summation and some changes of variables which didn't turn out to be very helpful. Any help regarding any techniques that could be used here or references to books/results that could be used for the simplification will be appreciated. Thanks in advance!