Question: How and why did Maple do this Sum correctly?

1. How did Maple come up with this answer?

I've tried all the packages in SumTools and none of them give an answer except Hypergeometric.

2. Is there a way to trace the steps Maple is using so I can try and answer this myself?

3. Why did it sum the series when I didn't even ask - I deliberately used the inert form (no arguments though - I like what it did).

Thank you.


H1a := Sum(GAMMA(2*b - 1 + 2*n)*GAMMA(2*n - s)*(b - 1/2 + 2*n)/(GAMMA(2*b + 2*n + s)*GAMMA(2*n + 1)), n = 0 .. infinity);


The answer H1b it came up with is 

GAMMA(-s)*2^(1 + 2*b)*GAMMA(b)/(8*GAMMA(b + s)*2^(2*b));

which seems to be correct.

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