Evaluating $\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$, where $a=-\sqrt3+\sqrt5+\sqrt7$ , $b=\sqrt3-\sqrt5+\sqrt7$, $c=\sqrt3+\sqrt5-\sqrt7$

Let $a=-\sqrt{3}+\sqrt{5}+\sqrt{7}$ , $b=\sqrt{3}-\sqrt{5}+\sqrt{7}$, $c=\sqrt{3}+\sqrt{5}-\sqrt{7}$. Evaluate: $$\sum_{cyc} \frac{a^4}{(a-b)(a-c)}$$

What I have tried so far is writing the denominators in terms of $(a+b+c)$ but that didn't help too much. I have also tried establishing a lower bound on the sum, but that won't work either.

Any help is appreciated!

Solution 1:

HINT:

Check first that the algebraic expression in $a$, $b$, $c$ simplifies to
$$ a^2 + b^2 + c^2 + a b + a c + b c$$ (there are many ways to get that, including just calculations)

Now reduce it to $$\frac{1}{2}((a+b)^2 + (a+c)^2 + (b+c)^2)$$

Now it should be simple. Note that the expression in $a$, $b$, $c$ is a Schur function in $a$, $b$, $c$, being $s_{(2,0,0)}(a,b,c)$.