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I am trying to get a simple but relatively ugly result out of the following summation, however, mathematica gives me an expression which includes HypergeometricPFQ. Obviously, I am doing something wrong, which is pretty classical for me. Can somebody please point me out what went wrong? The result should be an expression involving factorials and product (maybe). thank you!

a[j_, q_, k_] := (q! (j + q - 1)!)/((j! (q - j)!) (j - k + q - 1)!);
aa[i_, q_, k_] := a[i - q + k + 1, q, k];
b[j_, q_, k_] = (-1)^j Sum[aa[i, q, k]*aa[j - i, q, k], {i, 0, j, 1}]

((-1)^j k! (j + k)! (q!)^2 HypergeometricPFQ[{-j, 1 + k, 
 1 + k - 2 q, -1 - j - k + q}, {-j - k, 2 + k - q, -j - k + 2 q}, 
1])/(j! (1 + k - q)! (1 + j + k - q)! (-1 - k + 2 q)! (-1 - j - 
  k + 2 q)!)
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Did you mean to use = vs := in last definition? Fixing that and invoking b with some arguments results in numeric values... the "mess" you see is a result of using Set (=) - that's just the definition being spit back with mathematical jiggering applied. –  rasher Mar 21 at 2:24
    
I am not sure if that matters. I was hoping to get an expression for b[j,q,k], in terms of factorials and products, involving q, j and k, frankly. –  No Leaf Clover Mar 21 at 2:31
    
Take a look at (34) (34bis) and (35) here mathworld.wolfram.com/GeneralizedHypergeometricFunction.html .Perhaps some of them apply –  belisarius Mar 21 at 2:57
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