The following is a well-known Bessel function identity:
$$J_{-n}(z)=(-1)^n J_n(z),\qquad n\in\mathbb Z$$
To check this, I used the following code and the result is as what I expected.
In[2]:= FullSimplify[(-1)^n*BesselJ[n, z] == BesselJ[-n, z], n ∈ Integers]
Out[2]= True
The problem is that Mathematica does not return zero when I try to simplify the following expression:
$$(-1)^n J_{n}(z)-J_{-n}(z),\qquad n\in\mathbb Z$$
I tried the following code, but the output is as complex as the input:
In[3]:= FullSimplify[(-1)^n*BesselJ[n, z] - BesselJ[-n, z], n ∈ Integers]
Out[3]= -BesselJ[-n, z] + (-1)^n BesselJ[n, z] (*result I expected : 0*)
My goal is to command Mathematica to reduce the expression to zero, and I need some advice.