Partition identities and a continued fraction of Ramanujan
The authors explore identities for the numerator and denominator of Ramanujan's continued fraction: \[ R(a,b)=1+{bq \over \displaystyle 1+aq+ {bq^ 2 \over \displaystyle 1+aq^ 2+ {bq^ 3 \over \displaystyle 1+\cdots}}}. \] In particular, they find partition identities that generalize results of Sylvester, Göllnitz, and this reviewer. Representative of their results is the identity that the number of partitions of \(n+j\) into \(i+j\) distinct red parts and \(j\) distinct blue parts, each blue part being \(\leq i+j\), is equal to the number of partitions of \(n\) into \(i\) red parts and \(j\) blue parts such that all parts have distinct values and no part is one greater than any blue part. They also generalize the Odlyzko and Wilf result on the number of \(n\)- coin fountains with \(k\) coins at the base to fountains of two-colored coins.
scientific article; zbMATH DE number 217949
scientific article