Orthogonal Polynomials and Special Functions

Asymptotics of Jacobi matrices for a family of fractal measures GÖKALP APLAN BILKENT UNIVERSITY, TYRKEY There are many results concerning asymptotics of orthogonal polynomials and Jacobi matrices associated with a measure whose essential support is a finite union of intervals on R. In the case of totally disconnected support, spectral theory of orthogonal polynomials is less complete and numerical evaluations can be seen as a useful source of information. In this talk, we discuss various properties and asymptotics of Jacobi matrices for a special family of fractal measures. We focus on numerical results and conjectures. The talk is based on a joint work with Alexander Goncharov and Ahmet Nihat Şimşek. On the median of the beta distribution DIMITRIOS ASKITIS UNIVERSITY OF COPENHAGEN The median q of the beta distribution is defined implicitly by the equation ∫ q 0 ta−1(1− t)b−1dt = 1 2 ∫ 1 0 ta−1(1− t)b−1dt. We study the monotonicity and asymptotic properties of the median as a univariate function of the parameter a, as well as of the function a log q(a), related to its logarithm. In particular, we find asymptotic expansions for a → 0 and ∞. These are related to the polygamma function and generalised Bernoulli polynomials.

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