Recently there has been a great deal of interest in the theory of
orthogonal polynomials. The number of books treating the subject,
however, is limited. This monograph brings together some results
involving the asymptotic behaviour of orthogonal polynomials when
the degree tends to infinity, assuming only a basic knowledge of
real and complex analysis. An extensive treatment, starting with
special knowledge of the orthogonality measure, is given for
orthogonal polynomials on a compact set and on an unbounded set.
Another possible approach is to start from properties of the
coefficients in the three-term recurrence relation for orthogonal
polynomials. This is done using the methods of (discrete)
scattering theory. A new method, based on limit theorems in
probability theory, to obtain asymptotic formulas for some
polynomials is also given. Various consequences of all the results
are described and applications are given ranging from random
matrices and birth-death processes to discrete SchrAdinger
operators, illustrating the close interaction with different
branches of applied mathematics.
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