Decomposition of Laguerre polynomials with respect to the cyclic group of order \(n\) (Q1298549)

Here, the author considers a generalized case of Laguerre polynomial \(L^{(\alpha)}_m\), \[ L^{(\alpha,n,k)}_m(z)= {1\over n} \sum^{n- 1}_{\ell= 0} \exp\Biggl(-{2i\pi k\ell\over n}\Biggr) L^{(\alpha)}_m\Biggl(z\exp \Biggl({2i\pi\ell\over n}\Biggr)\Biggr),\quad z\in\mathbb{C} \] with two additional parameters \(n\) and \(k\). A close relationship between \(L^{(\alpha, n,k)}_m(z)\) and the Brafman polynomials is established. First, a derivation, from the basic identities and relations for \(L^{(\alpha)}_m\), other analogues for \(L^{(\alpha, n,k)}_m\) that turn out to be two integral representations, an operational representation, some generating functions defined by means of the generalized hyperbolic functions of order \(n\) and the hyper-Bessel functions, some finite sums including multiplication and addition formulas, a non-standard \((2n+1)\)-term recurrence relation and a differential equation of order \(2n\). Then some identities of \(L^{(\alpha)}_m\) are expressed as functions of the polynomials \(L^{(\alpha, n,k)}_m\). Finally, some properties of the first component \(L^{(\alpha, n,0)}_m\) are also given.