They are closely related to generalized hypergeometric series.
The integral is usually taken along a contour which is a deformation of the imaginary axis passing to the right of all poles of factors of the form Γ(a + s) and to the left of all poles of factors of the form Γ(a − s).
This equality can be obtained by moving the contour to the right while picking up the residues at s = 0, 1, 2, ... .
Given proper convergence conditions, one can relate more general Barnes' integrals and generalized hypergeometric functions pFq in a similar way (Slater 1966).
There are analogues of Barnes integrals for basic hypergeometric series, and many of the other results can also be extended to this case (Gasper & Rahman 2004, chapter 4).