Mathematica ruins domain after explicit summation?

Question

I'm working with this function

aExpl[G_,T_] := -2^(G - 2 T) Sum[(-1)^r (2 r - 1) Binomial[G,2 r] Sum[(-1/2)^j Binomial[r, G - T - j] Binomial[G - T - j,T], {j, G - T - r, G - 2 T}], {r, T, G/2}]

and I have no problems with it when written like this. Specifically, I get

aExpl[0,0] = 1
aExpl[1,0] = -1
aExpl[2,1] = 1

as you can check for yourself. I would like to have my sums evaluated and I tried to do that with and without specifying my conditions in more than one way. But whatever I do, the answer comes out as

((-1)^(1 + G - T) 2^(-1 + G) ((-1 + G - T) (-1 + 2 T) Binomial[G, 2 T] + (1 + T) (1 + 2 T) Binomial[G, 2 (1 + T)]) Gamma[-1 + G - T] Gamma[1/2 + T])/(Sqrt[\[Pi]] Gamma[G])

which obviously doesn't work for those three specific values. I tried redefining the function in parts, so that the above result holds for values that interest me and are defined (G>=2T && G>=3), along with setting problematic values explicitly.

aExpl[0,0]:=1
aExpl[1,0]:=-1
aExpl[2,1]:=1

This gives the correct result for all values, but the main reason why I have the function aExpl[G,T] in the first place is to sum it over Gs and Ts. But I can't do that, because Mathematica messes something up again. For Example aExpl[0,T] gives 0, which obviously isn't the case.

What in the world is going on here? :O

Answer 1

alYour result is actually a rather pathological function. The problematic term is the ratio of Gamma[G - T - 1]/Gamma[G] for integer arguments, which gives indeterminate (infinity/infinity) whenever (G == 0 && T [Element] Integer) and infinity (sometimes multiplied by zero from the rest of the result function) whenever (G, T [Element] Integer) && (G <= T + 1). You would have to set explicit values for all cases with G = 0 and all cases with (G, T [Element] Integer) && (G <= T + 1) that 'interest' you.

Edit: Now I think there is no real problem here. For all cases (apart from the three you quoted) the result is correct:

N[MatrixForm[
Table[{{G, T}, 
aExpl[G,T], (1/(Sqrt[Pi]*Gamma[G]))*(-1)^(1 + G - T)*
2^(-1 + G)*((-1 + G - T)*(-1 + 2*T)*Binomial[G, 2*T] + (1 + T)*(1 + 2*T)*
Binomial[G, 2*(1 + T)])*Gamma[-1 + G - T]*Gamma[1/2 + T]},
{G, 0, 7, 1}, {T, 0, G/2, 1}]]]

The three cases have the problem with the Gamma function I described above. And the case (G=0,T) is not covered by your condition (G>=2T && G>=3) and restricts the summation to zero.

Answer 2

Your summation function has certain peculiar behavior which needs to have special cases beyond the general cases. Your original function (with tweaks):

aExp1[G_Integer, T_Integer] := -2^(G - 2 T)
   Sum[(-1)^r (2 r - 1) Binomial[G, 2 r] 
      Sum[(-1/2)^j Binomial[r, G - T - j]
                   Binomial[G - T - j, T], 
   {j, G - T - r, G - 2 T}], {r, T, G/2}];

My version without explicit summation is the following:

aexp1[G_Integer, T_Integer] := If[T<0 || G<2 T,
   0, -(-1)^(G+T) 2^(G-2 T-2) If[G<=2, 2,
   If[T==3, 1/2, 1]] Which[T==0, G-2, T==1,
      G (G-1), True, G (G-1) (G-2) Product[
      G-T-i, {i, 2, T-1}] / 
   If[T<4, Max[1, T], T!]]];

My version returns 0 if T<0 or G<2*T while your version returns non zero results for those cases.