## Mathematics, Department of

#### Title

#### Date of this Version

1941

#### Abstract

The identity

(1) Σ_{n=1}^{∞} (*q ^{n}*)/[(1-

*q*)

^{n}^{2}] {(1÷(1-

*q*)) + (1÷(1-

*q*

^{2})) + … + (1 ÷ (1 -

*q*))} = Σ

^{n}_{n=1}

^{∞}[(

*n*

^{2}

*q*)

^{n}was deduced from arithmetical considerations by E. T. Bell. About five years ago, W. N. Bailey proved the relation

(2) Σ

_{n=0}

^{∞}[(1-

*q*)(1-

*q*

^{2})…(1-

*q*] ÷ [(1-

^{n}*z*)(1-

*qz*)…(1-

*q*)] x [(

^{n}z*z*+ 1) ÷ (1 -

^{n}*q*+1] = Σ

^{n}_{n=0}

^{∞}[(

*q*) ÷ (1-

^{n}z*q*)

^{n}z^{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.

## Comments

Published in

Bull. Amer. Math. Soc.47 (1941) 781-784.