Given the definition of the spherical Bessel function,
Prove the recurrence relation:
The Attempt at a Solution
The method to prove the recursion relation should be completed using proof by induction. This really comes down to the formalities involved in completing a proof of this nature: proof by induction implies that if the to-be-proved relation holds for one value (say, [itex]k[/itex]) then it may be induced that it holds for subsequent values ([itex]k+1[/itex]). My question: must I demonstrate that the relation holds for any arbitrary [itex]k[/itex], or can I just pick one (say, 1) and then show that the relation holds for what you would expect from the original equation (for 1+1=2)?