The generalization in the title is defined as the analytic continuation of the series
with appropriate restrictions on , , . The case was considered by E. W. Barnes in [Cambr. Trans. 19, 374–425 (1904; JFM 35.0462.01) and ibid. 19, 322–355 (1904; JFM 35.0462.02)]. Integral representations are obtained, together with a basic summation formula that expresses as a power series in whose th coefficient involves . This basic identity leads to interesting evaluations of classes of series associated with .