Formula for the reflection length of elements in the group G(m,p,n)

We deduce a closed formula for the reflection length functions on the reflection group G(m,p,n) with p [m] and pm. The formula is shown separately in three cases: p=1, p=m and p [2,m−1] (see the proof of Theorems 2.1, 3.1 and 4.4).