Summary: We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions \[ \ell^ a_ n(x)=(n!/\Gamma (n+a+1))^{1/2} e^{-x/2} L^ a_ n(x),\quad n=0,1,2,..., \] in \(L^ 2({\mathbb{R}}_+,x^ adx)\), \(a\geq 0\). We prove that the Cesàro means of order \(\delta >a+2/3\) of any function \(f\in L^ p(x^ adx),\quad 1\leq p\leq \infty,\) converge to f a.e. The main tool we use is a Hardy- Littlewood type maximal operator associated with a generalized Euclidean convolution.