In this paper, we define convolutions $( f\ast g( [ \alpha ] ) ) (z)$ and $( f\ast h( [ \alpha ] ) )(z)$ of functions analytic in the open unit disk with some non-zero parameter $\alpha $, satisfying certain recurring relations. Making use of admissible function method introduced by Miller and Mocanu, certain geometric properties of these convolutions are obtained. Taking specific forms of the functions $g( [ \alpha ] ) $ and $h( [\alpha ] ) $, some consequences of our results are also given.