How to show these two expressions are the same?

Question

I think

Can be simplified to  $-x^2$
I tried Reduce, FullSimplify, FunctionExpand, and Mathematica is not able to confirm this.
Although for $x>8$ or so, the res expressions do not evaluate numerically to same as $-x^2$. This could be due to numerical issue in evaluating res.
ClearAll[x];
res=1/4 Hypergeometric0F1[4/3,x^3/9] (-8 x^2 HypergeometricPFQ[{1/3},{2/3,4/3},x^3/9]+x^5 HypergeometricPFQ[{4/3},{2/3,7/3},x^3/9])+Hypergeometric0F1[2/3,x^3/9] (x^2 HypergeometricPFQ[{2/3},{4/3,5/3},x^3/9]-1/5 x^5 HypergeometricPFQ[{5/3},{4/3,8/3},x^3/9]);
f=-x^2;
Plot[{f,res},{x,-20,9},PlotStyle->{Red,Blue}]

Any suggestions?
Table[{f,res}/.x->n,{n,-2,2,.1}]

J. M.'s ennui · Answer

Here is an indirect way to prove this. Applying DifferentialRootReduce[] to your res gives a tenth-order linear ODE:

Short[ode = First[Head[DifferentialRootReduce[res, x]]][y, x][[1]]]
   (-8741760 - 1918960 x^3 + 143800 x^6 - 2560 x^9) y[x] +
   (-7160448 x + 2599248 x^4 - 337056 x^7 + 6144 x^10) y'[x] +
   (11531328 x^2 - 1639768 x^5 + 265156 x^8 - 4864 x^11) y''[x] + <<6>> +
   (22272 - 36800 x^3 + 17144 x^6 - 704 x^9) Derivative[9][y][x] +
   (-11136 x + 7360 x^4 - 2143 x^7 + 64 x^10) Derivative[10][y][x] == 0

which means a solution of this ODE such as yours would be uniquely determined by ten initial conditions. Then, look at the series expansion, whose coefficients are exactly what you'd supply as initial conditions:

Series[res, {x, 0, 10}]
   -x^2 + O[x]^11

and you should be able to take it from there.

How to show these two expressions are the same?

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