The centralizer of the Laguerre polynomial set (Q5902934)

The set \(\pi\) of all simple polynomial sets with umbral product is a noncommutative group with the identity \(I=\{x^ n,\quad n=0,1,2,...\}.\) The authors have studied the characterization of the elements of the centralizer \(C_{\pi}(L^{\alpha})\) of the Laguerre polynomial set in the group \(\pi\). Four special cases have also been considered. The interesting special case is the symmetric subgroup \(\Sigma\) of \(\pi\) and in this connection the authors have proved that \(C_{\Sigma}(L^{\alpha})\) is a commutative subgroup of \(C_{\pi}(L^{\alpha})\).