Here is an outline of a proof. The full argument is quite long.
Step 1. If then
Reason: Let . Using the relations and , you can check that This is equal to because
Step 2. If m, n are positive integers then
Reason: In Step 1, let . Then .
Step 3. For and , Also,
Reason: Follows immediately from Step 2 because .
Step 4. .
Reason: Just add up all the relations in Step 3.
Finally, divide by to get the result.