Decomposition of Laguerre polynomials with respect to the cyclic group of order n

Let n be an arbitrary positive integer, We decompose the Laguerre polynomials Lm(α) as the sum of n polynomials Lm(α,n,k); m ∈ N; k = 0, 1,…, n − 1; defined by Lm(x,n,k)(z)=1n∑t=0n−1exp−2iτklnLm(x)z exp2iτln,zϵC s paper, we establish the close relation between these components and the Brafman polynomials. The use of a technique described in an earlier work [2] leads us firstly to derive, from the basic identities and relations for Lm(α), other analogous for Lm(α,n,k) 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. Secondly, to express some identities of Lm(α) as functions of the polynomials Lm(α,n,k). Some particular properties of Lm(α,n,0), the first component, will be pointed out.