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Given $K$ arbitrary poles, which are neither necessarily distinct norbounded, on the extended real line, a corresponding ordered base of rationalfunctions orthogonal with respect to varying exponential weights isconstructed: this gives rise to a $K$-fold family of orthogonal rationalfunctions (ORFs). The ORF problem is characterised as a family of $K$ matrixRiemann-Hilbert problems (RHPs) on the extended real line, and a correspondingfamily of $K$ energy minimisation (variational) problems containing externalfields with singular points is formulated, and the existence, uniqueness, andregularity properties of the associated family of equilibrium measures isestablished. The family of $K$ equilibrium measures is used to derive a familyof $K$ model matrix RHPs on the extended real line that are amenable toasymptotic analysis via the Deift-Zhou non-linear steepest-descent method: thisis used to derive uniform asymptotics, in a certain double-scaling limit, ofthe ORFs and their leading coefficients, as well as related, important objects,in the entire complex plane. A family of $K$ multi-point Pad\'e approximants(MPAs) for the Markov-Stieltjes transform is also presented, and uniformasymptotics, in a certain double-scaling limit, are obtained for thecorresponding MPAs and their associated errors in approximation (MPA errorterms) in the entire complex plane.

Riemann-Hilbert Characterisation of Rational Functions with a General  Distribution of Poles on the Extended Real Line Orthogonal with Respect to  Varying Exponential Weights: Multi-Point Pad\'e Approximants and Asymptotics