Remarks on some partition identities (Q1196590)

The authors utilize some power series expansions involving infinite products to obtain some new transformation formulas. For example, they prove the following two main theorems: Theorem 1. Let \(| x|<1\), \(| q_ i|<1\) \((i=1,2,\ldots,k)\). Write \[ f(x)=f(x;q_ 1,\ldots,q_ k)=\sum^ \infty_{n=1}{x^ n\over n(1-q^ n_ 1)\ldots(1-q^ n_ k)}, \] \[ F(x)=F(x;q_ 1,\ldots,q_ k)=\prod^ \infty_{{n_ i=0\atop 1\leq i\leq k}}(1-xq_ 1^{n_ 1}\ldots q_ k^{n_ k})^{-1}. \] Then \(\log F(x)=f(x)\). Theorem 2. If \(| x|<1\), \(| q|<1\), then \[ \prod^ \infty_{n=1}(1-xq^ n)^{-n}=\exp\left(\sum^ \infty_{n=1}{x^ nq^ n\over n(1-q^ n)^ 2}\right). \] By using these transformation formulas some new partition theorems are derived.