On Certain Basic Series

The identity (1) Σn=1∞ (qn)/[(1-qn)2] {(1÷(1-q)) + (1÷(1-q2)) + … + (1 ÷ (1 -qn))} = Σn=1∞ [(n2qn) was deduced from arithmetical considerations by E. T. Bell. About five years ago, W. N. Bailey proved the relation (2) Σn=0∞ [(1-q)(1-q2)…(1-qn] ÷ [(1-z)(1-qz)…(1-qnz)] x [(zn + 1) ÷ (1 - qn +1] = Σn=0∞ [(qnz) ÷ (1-qnz)2], from which he obtained (1) by differentiating with respect to z and then putting z = q. A short time later Hall gave an alternate proof of (2) by simply specializing the parameters in a relation between basic series