International Journal of Mathematics and Mathematical Sciences
Volume 17 (1994), Issue 2, Pages 369-378

A generalized formula of Hardy

Geoffrey B. Campbell

Institute of Advanced Studies, School of Mathematical Sciences, The Australian National University, GPO Box 4, Canberra 2601, Australia

Received 16 November 1992

Copyright © 1994 Geoffrey B. Campbell. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We give new formulae applicable to the theory of partitions. Recent work suggests they also relate to quasi-crystal structure and self-similarity. Other recent work has given continued fractions for the type of functions herein. Hardy originally gave such formulae as ours in early work on gap power series which led to his and Littlewood's “High Indices” Theorem. Over a decade ago, Mahler and then others proved results on irrationality of decimal fractions applicable to types of functions we consider.