You're almost there! Notice that . So your expression for is equal to , as required.

Note 1. Instead of starting with the desired formula for and then showing that it is equal to , it is more usual when writing out a proof by induction to reverse that process. In other words, start with the formula (*) for and then show that is equal to the desired formula for .

Note 2. Laguerre's name has an e after the u.