Let n be an arbitrary positive integer, We decompose the Laguerre polynomials L ( α ) m as the sum of n polynomials L ( α , n , k ) m ; m N; k = 0, 1,..., n - 1; defined by L ( α , n , k ) m (z) = 1n l=0n-1 exp (- 2iπkln) L ( α ) m (z exp(2iπln)), z C.In this paper, we establish the close relation between these components and the Brafman polynomials. The use of a technique described in an earlier work leads us firstly to derive, from the basic identities and relations for L ( α ) m , other analogous for L ( α , 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. Secondly, to express some identities of L ( α ) m as functions of the polynomials L ( α , n , k ) m . Some particular properties of L ( α , n , 0 ) m , the first component, will be pointed out.