Series solutions to the TOV equations

We present general series solutions to the Tolman-Oppenheimer-Volkoff equations for compact stellar objects. We develop an algorithm to compute the coefficients of the power series in terms of the equation of state and its derivatives with respect to the thermodynamic variables. Using these results, we establish general properties of analytic solutions and their relation to the regularity of the equation of state. Applying the theory of Pad\'e approximants, we derive series representations for meromorphic functions whose domains of convergence may include isolated poles. These analytic solutions are then used to obtain closed-form expressions to approximate the radius and mass of stellar objects. We apply the formalism to specific models, namely fluids with affine equations of state and polytropic fluids, and compare the results with those obtained from numerical integration. Lastly, we extend the formalism to piecewise equations of state, deriving series solutions that can be matched across transition hypersurfaces.