Summary: The purpose of this paper is to establish oscillation criteria for the second order nonlinear dynamic equation
on an arbitrary time scale , where is a quotient of odd positive integers and is a positive -continuous function on . The function satisfies and and . We establish some new sufficient conditions under which the above equation is oscillatory by using the generalized Riccati transformation. Our results in the special cases when and involve and improve some oscillation results for second-order differential and difference equations; and when , and our oscillation results are essentially new. Some examples illustrating the importance of our results are included.