Spiridonov, V.P.; Zhedanov, A.S. Generalized eigenvalue problem and a new family of rational functions biorthogonal on elliptic grids. (English) Bustoz, Joaquin (ed.) et al., Special functions 2000: current perspective and future directions. Proceedings of the NATO Advanced Study Institute, Tempe, AZ, USA, May 29-June 9, 2000. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 30, 365-388 (2001). [ISBN 0-7923-7119-4/hbk] This is a survey of results by these authors published elsewhere [A. Zhedanov, “Biorthogonal rational functions and the generalized eigenvalue problem”, J. Approximation Theory 101, No.2, 303-329 (1999; Zbl pre01398588)] [V. Spiridonov and A. Zhedanov, “Spectral transformation chains and some new biorthogonal rational functions”, Commun. Math. Phys. 210, No. 1, 49-83 (2000; Zbl 0989.33008); “Classical biorthogonal rational functions on elliptic grids”, C. R. Math. Acad. Sci., Soc. R. Can. 22, No. 2, 70-76 (2000; Zbl 0974.33016)] [A. S. Zhedanov and V. P. Spiridonov, “Hypergeometric biorthogonal rational functions”, Russ. Math. Surv. 54, No. 2, 461-463 (1999); translation from Usp. Mat. Nauk 54, No. 2, 173-174 (1999; Zbl 0940.33003)] and a forthcoming paper. The idea is the following: for orthogonal polynomials the 3-term recurrence relation can be written as JP(x) = xP(x) where J is a tridiagonal Jacobi matrix and P(x) the vector of orthogonal polynomials. Given two Jacobi matrices, the generalized eigenvalue problem J1R(z) = zJ2R(z) gives a vector R(z) of rational functions that are biorthogonal with respect to summation over a set of points on an elliptic grid (it can be generated by an elliptic function). Related material is generalized correspondingly. For example, elliptic analogues of the very-well-poised balanced hypergeometric functions are defined and their properties are formulated. A class of rational functions is described that can be expressed in terms of these elliptic analogues and that satisfy 3-term recurrence relations. Properties about their zeros, their orthogonality, a difference equation they satisfy, properties of the grid points, and associated continued fractions are discussed. Adhemar Bultheel (Leuven) Keywords : three-term recurrence relation; biorthogonal rational functions; orthogonal polynomials; elliptic functions; hypergeometric functions Classification : 42C05 General theory of orthogonal functions and polynomials 33C45 Orthogonal polynomials and functions of hypergeometric type 39A70 Difference operators