Final Proof of the Identity and of the Second Order Recurrence
Let be any solution of the second order recurrence which has been obtained for the left-hand side:
Then, also solves the recurrence for the right-hand side:
At this point, we have proved that both sides of the equation satisfy the same recurrence of order 4. To prove the announced equality, we simply need to check 4 initial conditions, since the leading coefficient of the recurrence of order 4,
never vanishes for non-negative . Now the proof of the identity
simplify follows from
Therefore, the Apéry numbers also satisfy the announced second order recurrence.