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The paper undertakes certain special forms of the quarter symmetric metric and non-metric connections on an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula>-anti-Kähler manifold. Firstly, we deduce the relation between the Riemannian connection and the special forms of the quarter symmetric metric and non-metric connections. Then, we present some results concerning the torsion tensors of these connections. In addition, we find the forms of the curvature tensor, the Ricci curvature tensor and scalar curvature of such connections and we search the conditions for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula>-anti-Kähler manifold to be an Einstein space with respect to these connections. Finally, we study <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>R</mi><mi>i</mi><mi>c</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-vector fields with respect to these connections and give some results related to them.