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Following previous work of ours in spherical symmetry, we here propose a newparametric framework to describe the spacetime of axisymmetric black holes ingeneric metric theories of gravity. In this case, the metric components arefunctions of both the radial and the polar angular coordinates, forcing adouble expansion to obtain a generic axisymmetric metric expression. Inparticular, we use a continued-fraction expansion in terms of a compactifiedradial coordinate to express the radial dependence, while we exploit a Taylorexpansion in terms of the cosine of the polar angle for the polar dependence.These choices lead to a superior convergence in the radial direction and to anexact limit on the equatorial plane. As a validation of our approach, we buildparametrized representations of Kerr, rotating dilaton, andEinstein-dilaton-Gauss-Bonnet black holes. The match is already very good atlowest order in the expansion and improves as new orders are added. We expect asimilar behavior for any stationary and axisymmetric black-hole metric.

General parametrization of axisymmetric black holes in metric theories of gravity