Continuous representations of the meta Hahn algebra and special functions

Infinite families of biorthogonal rational functions are constructed using infinite dimensional representations of the three-generated meta Hahn algebra. Continuous Hahn and continuous dual Hahn polynomials are also recovered along the way using an embedding of the Hahn algebra inside the meta Hahn. Rational functions and polynomials arise as transition coefficients between eigenvectors and generalized eigenvectors, and their algebraic properties follow naturally from the relations of the meta Hahn algebra. Finally, biorthogonality relations are established for the rational functions introduced in the paper.