A Carlitz identity for the wreath product Cr≀Sn

We present in this work a new flag major index fmajr for the wreath product Gr,n=Cr≀Sn, where Cr is the cyclic group of order r and Sn is the symmetric group on n letters. We prove that fmajr is equidistributed with the length function on Gr,n and that the generating function of the pair (desr,fmajr) over Gr,n, where desr 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.