Generalisation is a way of taking a standard result and removing some of the conditions without making the proof untrue. A well-known example of this is the area of a rectangle. Everyone knows this is equal to height x width .
Specialisation has two possible purposes.
However, it is also possible to find new theorems for subsets of special cases. In this way, specialisation is forming new supporting evidence because it is leading to suggestions of a rather different result. Examples include any mathematical modelling problem - the formula is meant to be an analogy of the physical case. In the book Proofs and Refutations, author Imre Lakatos uses a conversational style between several imaginary pupils each offering a different argument towards a conjecture. If you try some examples yourself, you'll find that this does seem to hold very well as a rule.