# Reduce a MeijerG to elementary function

The following function for $$t>0$$

f[t_]:=1/(8 (-1+E^(2 t)) \[Pi]^(3/2)) E^t (MeijerG[{{1/2,1,3/2},{}},{{3/2},{}},-((2 I)/(5 t)),1/2]+MeijerG[{{1/2,1,3/2},{}},{{3/2},{}},(2 I)/(5 t),1/2]-MeijerG[{{1/2,1,3/2},{}},{{3/2},{}},-((2 I)/(3 t)),1/2]-MeijerG[{{1/2,1,3/2},{}},{{3/2},{}},(2 I)/(3 t),1/2])


is actually an elementary function, by

Plot[Evaluate[4Exp[4t]f[t]],{t,0,1},WorkingPrecision->20] but FunctionExpand can not simplify it,

FunctionExpand[f[t],Assumptions->t>0] Mathematica will automatically abort the evaluation. Using

AbortProtect@Trace@FunctionExpand[f[t],Assumptions->t>0]


we can trace the intermediate steps, but I didn't get some useful information from that.

Is there some other way to deal with MeijerG functions (if not for Mathematica, is there some useful Python package for this?)

It looks like Mathematica is not trying any of the known transformations of the Meijer G-functions.

We can use the second formula on this Wolfram Functions page,

f[t_] = 1/(8 (-1 + E^(2 t)) π^(3/2)) E^t (
MeijerG[{{1/2, 1, 3/2}, {}}, {{3/2}, {}}, -((2 I)/(5 t)), 1/2] +
MeijerG[{{1/2, 1, 3/2}, {}}, {{3/2}, {}}, (2 I)/(5 t), 1/2] -
MeijerG[{{1/2, 1, 3/2}, {}}, {{3/2}, {}}, -((2 I)/(3 t)), 1/2] -
MeijerG[{{1/2, 1, 3/2}, {}}, {{3/2}, {}}, (2 I)/(3 t), 1/2]);

f[t] /. MeijerG[a_, b_, z_, r_] :> MeijerG[1 - b, 1 - a, 1/z, r] //
FullSimplify[#, t > 0] &

(*    E^(-4 t)/4    *)

• Note that Entity["MathematicalFunction", "MeijerG"]["Dataset"] contains no entries, not even a link to the Wolfram Functions Site. Mar 23 at 19:08
• @BobHanlon A bug? Can you populate it? Strange, there is even Meijer GRepresentation row. Mar 25 at 19:18
• @ВалерийЗаподовников - only Wolfram can populate their database. Mar 25 at 19:50