# FullSimplify doesn't work in one case

We have the following code:

FullSimplify[
2/33 (-11 HypergeometricPFQ[{5/6}, {3/2, 11/6}, -(x^2/4)] +
x^2 HypergeometricPFQ[{11/6}, {5/2, 17/6}, -(x^2/4)] + (
11 Sin[x])/x)]


It doesn't work, but result is known, as it should be equal to zero. However, if we reformulate this problem as:

FullSimplify[
2/33 (-11 HypergeometricPFQ[{5/6}, {3/2, 11/6}, -(x^2/4)] +
x^2 HypergeometricPFQ[{11/6}, {5/2, 17/6}, -(x^2/4)] + (
11 Sin[x])/x) == 0](*True*)


We get the right solution.

What is the reason of this strange behaviour?

• This is not s strange behavior.It's normal behavior,because MMA dosen't know the answer.FullSimplify have some limitations, it's not a perfect function, but it's the best I know. – Mariusz Iwaniuk Jun 14 '18 at 7:39
• Quite likely, deciding wether an equation expr == 0 holds true is much easier than calculating the value of expr without any hints. Note that Simplify and FullSimplify have to chose among a bazillion of possibilities to simplify an expression. The information that it could be equal to 0 provides a valuable information. For example, it suffices to compare the Taylor series of the analytical expression at x = 0 to see that it is actually equal to 0. This is usually not so helpful for simplifying an expression. – Henrik Schumacher Jun 14 '18 at 7:49
• @HenrikSchumacher hm, it is really interesting information about FullSimplify. Thanks! – Андрей Кротких Jun 14 '18 at 8:26
• Many years ago someone wrote, I cannot remember who to give credit to, that people writing computer algebra software spend a lot of time thinking about expr==0 and how to do that really really well. But "simplify this" can mean many things and might not include the particular code which would recognize your problem and simplify your expression to zero. Perhaps that explains what you found. – Bill Jun 14 '18 at 18:21
• Series[expr, {x, 0, 100}] would suggest trying FullSimplify[expr == 0] – Bob Hanlon Jun 14 '18 at 19:31

As already noted, it is simply possible that FullSimplify[]'s (and even FunctionExpand[]'s) collection of transformations are not enough to deal with this problem. (See this for another example.)
In this case, however, one can enlist DifferentialRootReduce[] to prove what is needed:
DifferentialRootReduce[2/33 (-11 HypergeometricPFQ[{5/6}, {3/2, 11/6}, -(x^2/4)] +