Can Maple do these? Mma 9.0 and Macsyma can't. Not even half the special cases. Out[1015]= PolyGamma[n, 1/8] + (-1)^n PolyGamma[n, 3/8] == 2^(1 + n) PolyGamma[n, 1/4] - (2 I π)^(1 + n) PolyLog[-n, -((1 + I)/Sqrt[2])] n>0. In[1016]:= Table[%, {n, 0, 4}] In[1017]:= FullSimplify[%] Out[1017]= {False, 32 Catalan + 2 Sqrt[2] π^2 + PolyGamma[1, 3/8] == PolyGamma[1, 1/8], 0 == 12 Sqrt[2] π^3 + PolyGamma[2, 1/8] + PolyGamma[2, 3/8] + 448 Zeta[3], 8 (-16 + 11 Sqrt[2]) π^4 + 16 PolyGamma[3, 1/4] + PolyGamma[3, 3/8] == PolyGamma[3, 1/8], 0 == 912 Sqrt[2] π^5 + PolyGamma[4, 1/8] + PolyGamma[4, 3/8] + 380928 Zeta[5]} Out[1024]= (-1 + 2^(-1 - n)) PolyGamma[n, 1/6] + ((-1)^n 2^(-1 - n) + 2^(1 + n)) PolyGamma[n, 1/3] == I^(-1 + n) π^(1 + n) PolyLog[-n, 1/2 (-1 - I Sqrt[3])] - (-1)^ n (-1 + 3^(1 + n)) n! Zeta[1 + n] n>0. In[1025]:= Table[%, {n, 0, 4}] In[1026]:= FullSimplify[%] Out[1026]= {False, 4 π^2 + 3 PolyGamma[1, 1/6] == 15 PolyGamma[1, 1/3], True, 16 π^4 + 3 PolyGamma[3, 1/6] == 51 PolyGamma[3, 1/3], True} --rwg