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.