A Carlitz identity for the wreath product C_rn

We present in this work a new flag major index fmaj_r for the wreath product G_r,_n= C_rn, where C_r is the cyclic group of order r and G_n is the symmetric group on n letters. We prove that fmaj_r is equidistributed with the length function on Gr,_n and that the generating function of the pair (des_r,fmaj_r) over Gr,_n, where des_r is the usual descent number on Gr,_n, satisfies a "natural" Carlitz identity, thus unifying and generalizing earlier results due to Carlitz (in the type A case), and Chow and Gessel (in the type B case). A q-Worpitzky identity, a convolution-type recurrence and a q-Frobenius formula are also presented, with combinatorial interpretation given to the expansion coefficients of the latter formula.