# Is InverseMellinTransform unaware of second Barnes lemma?

Consider evaluating

res = InverseMellinTransform[( Gamma[a + s] Gamma[b + s] Gamma[c + s] Gamma[1 - d - s] Gamma[-s])/ Gamma[a + b + c - d + 1 + s], s, 1] // FunctionExpand


Even after the FunctionExpand the above returned hypergeometric functions instead of the expected gamma functions per second Barnes Lemma.

Numerically the result agrees

num = {a -> RandomComplex[WorkingPrecision -> 30],
b -> RandomComplex[WorkingPrecision -> 30],
c -> RandomComplex[WorkingPrecision -> 30],
d -> RandomComplex[WorkingPrecision -> 30]};
res - (Gamma[a] Gamma[b] Gamma[c] Gamma[1 - d + a] Gamma[ 1 - d + b] Gamma[1 - d + c])/( Gamma[e - a] Gamma[e - b] Gamma[e - c]) /. e -> a + b + c - d + 1 /. num


0.*10^-26 + 0.*10^-26 I

Is this expected behavior? If yes, this would suggest that InverseMellinTransform might return an unnecessarily complicated output in other cases as well. Perhaps there is some way to extract the simpler gamma function result from InverseMellinTransform which I am not aware of?

• My attitude is that Mathematica knows everything, and when it gives suboptimal results, it's just trolling you.... or at least that's how I feel. – QuantumDot Jun 19 '18 at 1:07