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

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

Michael 222 Maple 2019
sum
February 06 2021
0

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.

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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);

H1b:=convert(H1a,Hypergeom);

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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