On residually finite groups satisfying an Engel type identity

Let n, q be positive integers. We show that if G is a finitely generated residually finite group satisfying the identity [x,nyq]≡1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[x,_ny^q]\equiv 1$$\end{document}, then there exists a function f(n) such that G has a nilpotent subgroup of finite index of class at most f(n). We also extend this result to locally graded groups.