A further investigation of a pair of series transformation formulas with applications

As a theoretical completion or a substantial supplement to a recent joint paper by He et al. Discrete Math. 308 (2008), pp. 3427–3440] containing a pair of series transformation formulas with a variety of illustrative examples, we provide some convergence theorems for the transformation formulas under certain general conditions. We also show that these two transformation formulas subject to the convergence conditions can be further utilized to produce more than 30 special power series sums and combinatorial identities (in a wider sense) mostly not given previously.