Why FullSimplify doesn't work here?

Question

Since the emphasis of this question is on finding a workaround, I decided to post this question with an emphasis on the explanation of the behavior of Mathematica.

The Bessel function satisfies the following identity:

$$J_{-n}(z)=(-1)^n J_n(z),\qquad n\in\mathbb Z$$

Accordingly:

In[2]:= FullSimplify[(-1)^n*BesselJ[n, z] == BesselJ[-n, z], n ∈ Integers]
Out[2]= True

However:

In[3]:= FullSimplify[(-1)^n*BesselJ[n, z] - BesselJ[-n, z], n ∈ Integers]
Out[3]= -BesselJ[-n, z] + (-1)^n BesselJ[n, z]    (*result expected : 0*)

doesn't work as expected.

Some workarounds were suggested in the previous question, such as using ToString as the ComplexityFunction (in this answer):

FullSimplify[(-1)^n*BesselJ[n, z] - BesselJ[-n, z],  n ∈ Integers, 
             ComplexityFunction -> (StringLength @ ToString @ # &)]

My question is: Why FullSimplify with no specified option value for ComplexityFunction (thus using the default LeafCount) doesn't simplify $J_{-n}(z) - (-1)^n J_n(z)$ to zero (with $n\in\mathbb Z$)?

Answer 1

The problem is that though FullSimplify does its best to find as simple form as possible in a reasonable time, it's not smart enough to handle all possible cases (and, actually, that would be an undecidable problem in general).

But your indentity seems to be well-known and compelling, so I would recommend to send it as a suggestion to [email protected].