Multiple Orthogonal Polynomials, Irrationality and Transcendence

. We explain the notion of multiple orthogonal polynomials (polyorthogonal polynomials), which is closely related to Hermite-Pad'e rational approximation of a system of Markov functions, and give some explicit examples. As an application we show how multiple orthogonal polynomials can be used to give a constructive proof of irrationality of certain real numbers and also of transcendence of real numbers. Historically Hermite-Pad'e approximation was introduced by Hermite to prove the transcendence of e. 1. Orthogonal polynomials The notion of orthogonal polynomials is an old one, going back to the previous century (Chebyshev, Stieltjes). A very good source of information is Szego's book [Sz] and a more recent exposition can be found in [VA1]. Monic orthogonal polynomials for a positive Borel measure ¯ on the real line are polynomials Pn (n = 0; 1; : : : ) of degree n and leading coefficient one such that Z Pn (x)x k d¯(x) = 0; k = 0; 1; : : : ; n \Gamma 1: (1.1) These n orthogonali..