Abel-type weighted means transformations into \(\ell\) (Q1567233)

Let \(q_k ={k+\alpha\choose k}\) for \(\alpha>-1\) and \(Q= \sum^n_{k= 0} q_k\). Then the Abel-type weighted mean matrix denoted by \(A= (a_{nk})\) is defined by \(a_{nk}= q_k/Q_k\) if \(0\leq k\leq n\), and \(a_{nk}= 0\) if \(k> n\). It is regular. In this paper the authors study these transformations as mappings into \(\ell\) and prove necessary and sufficient conditions for \(A\) to be 1-1. Further they prove eight theorems showing some other properties of the matrix \(A\).