First of all, why, to save your life, don't the minus signs cancel in In[444]:= Simplify[FullSimplify[FunctionExpand[ -Sum[-((Gamma[-(1/4) + (5*n)/4]*y[n])/ (4*Gamma[3/4 + n/4]*Gamma[1 + n])), {n, 0, Infinity}]]]] Out[444]= -Sum[-((Gamma[-(1/4) + (5*n)/4]*y[n])/ (4*Gamma[3/4 + n/4]*Gamma[1 + n])), {n, 0, Infinity}] Why don't they cancel automatically? Now take two terms of the series In[445]:= % /. ∞-> 1 Out[445]= (Gamma[-(1/4)] y[0])/(4 Gamma[3/4]) + y[1]/4 observing that the constant (n=0) term is -1 if y[n_]:=y^n In[446]:= FunctionExpand[% /. y[n_] -> y^n] Out[446]= -1 + y/4 Now impose y[n_]:=y^n on the whole sum: In[447]:= %%% /. y[n_] -> y^n Out[447]= (1/(32 Gamma[3/4]))(8 Gamma[-(1/4)] + 8 Gamma[-(1/4)] HypergeometricPFQ[{-(1/20), 3/20, 7/20, 11/20}, {1/ 4, 1/2, 3/4}, (3125 y^4)/256] + 8 y Gamma[3/ 4] HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5}, {1/2, 3/4, 5/4}, ( 3125 y^4)/256] + 5 y^2 Gamma[3/ 4] HypergeometricPFQ[{9/20, 13/20, 17/20, 21/20}, {3/4, 5/4, 3/ 2}, (3125 y^4)/256] + 5 y^3 Gamma[3/ 4] HypergeometricPFQ[{7/10, 9/10, 11/10, 13/10}, {5/4, 3/2, 7/ 4}, (3125 y^4)/256]) This outrageously stupid and unhelpful canonicalization is what drove me to use y[n] when I meant y^n in the first place. This should never be automatic-- either put it on (overloaded) FunctionExpand, or provide ToHypergeometric, which surely is already present internally. And yet in %445 you won't canonicalize Gamma[-1/4]/Gamma[3/4], which would actually be helpful. But the most wonderful thing about the preceding eructile dysfunction is that it's wrong! In[448]:= FunctionExpand[% /. y -> 0] Out[448]= -2 should be -1 ! --Bill Gosper In[61]:= $LicenseID Out[61]= "L3290-7570"