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Abstract The asymptotic theory of orthogonal polynomials has a long history. Many of the early contributions are due to Szegö. In the 1980's the subject rapidly developed with the works of (among others) Nevai, Mhaskar, Saff, Lubinsky, Rakhmanov, Van Assche, and Totik. Recently, a new technique for computing asymptotic properties of orthogonal polynomials was created by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. Starting from the Fokas-Its-Kitaev characterization of orthogonal polynomials in terms of a Riemann-Hilbert problem and using the steepest descent technique that was previously developed for the asymptotic analysis of integrable systems, these authors obtained remarkably sharp results for orthogonal polynomials on the real line with respect to a varying exponential weight. These results came as a big surprise to the old community in orthogonal polynomials, and one is still in the process to understand the full potential of the new method.
In this series of lectures, I intend to introduce the Riemann-Hilbert method based on two kinds of problems. The first deals with polynomials that are orthogonal on the interval [-1,1] with respect to a Jacobi weight times an analytic factor. In the steepest descent method we will meet the Szego function associated with the weight and Bessel functions for the local analysis.
The second problem deals with Laguerre polynomials with large negative parameters. These polynomials are not orthogonal on the real line. However the Fokas-Its-Kitaev characterization continues to hold on certain contours in the complex plane. A new phenomenon is the selection of the right contour. The local analysis involves Airy functions, and in the case of coalescing turning points, parabolic cylinder functions.
Abstract In a series of lectures, a self-contained introduction will be given to the theory of 3nj-coefficients of su(2) and su(1,1), their hypergeometric expressions, and their relations to orthogonal polynomials. The 3nj-coefficients of su(2) play a crucial role in various physical applications (dealing with the quantization of angular momentum), but here we shall deal with their mathematical importance only.
I shall begin with a brief explanation of representations of the Lie algebra su(2), and of the tensor product decomposition. All this will be derived in a purely algebraic context (avoiding the notion of Lie groups for those that are not familiar with this). The Clebsch-Gordan coefficients (or 3j-coefficients) will be defined, and their expression as a hypergeometric series will be deduced. The relation with (discrete) Hahn polynomials will be emphasized.
The tensor product of three representations will be discussed, and the relevant Racah coefficients (or 6j-coefficients) will be defined. The explicit expression of a Racah coefficient as a hypergeometric series of 4F3-type will be derived. The connection with Racah polynomials and their orthogonality will be determined.
As a second example closely related to su(2), a class of representations (together with their 3j- and 6j-coefficients) of the Lie algebra su(1,1) will be considered.
Finally, we shall introduce general coupling theory: the tensor product of (n+1) representations is considered, and "generalized recoupling coefficients" or 3nj-coefficients are defined. In this context, the Biedenharn-Elliott identity is shown to play a crucial role. We shall derive various results from this identity, e.g. convolution theorems for certain orthogonal polynomials, orthogonal polynomials in several variables, and interpretations of 3nj-coefficients as connection coefficients.
Abstract Recently, there has been a surge of practical and theoretical interest on the part of mathematical physicists, classical analysts and abstract analysts in the subject of exponential asymptotics, or hyperasymptotics, by which is meant asymptotic approximations in which the error terms are relatively exponentially small. Such approximations generally yield much greater accuracy than classical asymptotic expansions of Poincaré type, for which the error terms are algebraically small: in other words, they lead to "exponential improvement." They also enjoy greater regions of validity and yield a deeper understanding of other aspects of asymptotic analysis, including the Stokes phenomenon.
I shall show how one can obtain readily-applicable theories of hyperasymptotic expansions of solutions of differential equations and for integrals with one or more saddles. The main tool will be the Borel transform, which transforms the divergent asymptotic expansions into convergent series. Other methods will also be mentioned.
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