Richard Askey

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.

A Set of Orthogonal Polynomials That Generalize the Racah Coefficients or

A very general set of orthogonal polynomials with five free parameters is given explicitly, the orthogonality relation is proved and the three term recurrence relation is found.

6 - j

F. H. Jackson defined q-analogue of the gamma function which extends the q-factorial . This function is examined and analogues of many of the classical facts about the gamma function are obtained. These include an analogue of the Bohr-Mollerup theorem, an asymptotic formula for large x, the duplication formula, and two connections with sums that approximate beta function integrals. The behavior of this function as q changes is also considered

Symbols

Some old polynomials of L. J. Rogers are orthogonal. Their weight function is given. The connection coefficient problem, which Rogers solved by guessing the formula and proving it by induction, is derived in a natural way and some other formulas are obtained. These polynomials generalize zonal spherical harmonics on spheres and include as special cases polynomials that are spherical functions on rank one spaces over reductive p-adic groups. A limiting case contains some Jacobi polynomials studied by Hylleraas that arose in work on the Yukawa potential.

The q-Gamma and q-Beta Functions†

Abstract An integral for [P n (α + μ,β) (x)] [P n (α + μ,β) (1)] in terms of [P n (α,β) (y)] [P n (α,β) (1)] with a positive kernel is obtained. For β = ± 1 2 this integral is equivalent to an important integral of Feldheim and Vilenkin connecting ultraspherical polynomials. As an application we show that P n (α,α) (x) P n (α,α) (1) = ∫ −1 1 P n (β,β) (y) P n (β,β) (1) dμ(y) where α > β ⩾ − 1 2 , − 1 ⩽ x ⩽ 1, and dμ(y) is a positive measure which depends on x but not n . For β = − 1 2 this is a result of Seidel and Szasz. Similar results are obtained for Jacobi polynomials and the positivity of certain sums of ultraspherical and Jacobi polynomials is obtained.

A Generalization of Ultraspherical Polynomials.

The theory of partitions has long been associated with so called basic hypergeometric functions or Eulerian series. We begin with discussion of some of the lesser known identities of L.J. Rogers which have interesting interpretations in the theory of partitions. Illustrations are given for the numerous ways partition studies lead to Eulerian series. The main portion of our work is primarily an introduction to recent work on orthogonal polynomials defined by basic hypergeometric series and to the applications that can be made of these results to the theory of partitions. Perhaps it is most interesting to note that we deduce the Rogers-Ramanujan identities from our solution to the connection coefficient problem for the little q-Jacobi polynomials.

Integral representations for Jacobi polynomials and some applications

A. Selberg evaluated an important multivariable extension of the beta function integral. Andrews found a related integral and evaluated it using a result of Dyson, Gunson and Wilson. Basic hypergeometric, or q-series, extensions of these integrals are considered and evaluated in the two-dimensional case. Conjectures are given for the values of these integrals in the n-dimensional case.

Enumeration of Partitions: The Role of Eulerian Series and q-Orthogonal Polynomials

Gangolli [6] discovered this convolution structure for special values of ac and J3 namely /3= 1/2, a = (n -1)/2; 3=0, ac n; and /3=1, a-2n + 1; 1 G= 3, c = 7. n here is a non-negative integer. Let P (a,0) (x) be the Jacobi polynomial of degree n, order (2,/) defined by P(?()are orthogonal onl (-1, 1) writh resplect tO ( 1 0x)at(1 ? x7)~ a ndl

Some Basic Hypergeometric Extensions of Integrals of Selberg and Andrews

Explicit orthogonality relations are found for the associated Laguerre and Hermite polynomials. One consequence is the construction of the [n − 1/n] Pade approximation to Ψ(a + 1, b; x)/Ψ(a, b; x), where Ψ(a, b; x) is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity.

A CONVOLUTION STRUCTURE FOR JACOBI SERIES.

In earlier papers, see [3] and its bibliography, some classical inequalities for polynomials and trigonometric polynomials were rewritten as inequalities for Jacobi polynomials, and these inequalities suggested other inequalities, some of which have been proved while others are still just conjectures.