In this letter we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, An(z) and Bn(z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w0 and a standard jump function, then An(z) and Bn(z) have apparent simple poles at these jumps. We exemplify the approach by takingw0 to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and offdiagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painleve IV.

In this paper, we study those polynomials, orthogonal with respect to a particular weight, over the unionin of disjoint intervals, first introduced by N.I. Akhiezer, via a reformulation as a matrix factorization or Riemann-Hilbert problem. This approach complements the method proposed in a proevious paper, that involves the construciton of a certain meromorphic function on a hyperelliptic Riemann surface. The method de scribed here is based on the general Riemann-Hilert scheme of the theory of integrable systems and will enable us to derive, in a versy strightforward way, the relevant system of Fuchsian differential equations for the polynomials and the associated system of the Schlesinger deformation equations for certain quantaties involing the corresponding recurrence coffcients. Both of these equations wre obtained earlier by A. Magnus. In our aproach, however, we are able to go beyond Magnus's results by actualy solving the equatins in terms of the Riemann Θ-funcitons. We also show that the related Hankel determinant can be nterpreted as the relevantτ-function.

In this paper we show, how a straightforward and natural application of a pair of fundamental identities valid for polynomials orthogonal over the unit circle, can be used to calculate the determinant of the finite Toeplitz matrix, △n=det(wj−k)n−1 j,k=0:=det(∫|z|=1 w(z) zj−k+1 dz 2πi n−1 j,k=0, with the Fisher-Hartwig symbol, w(z)=C(1− z) +iβ(1− 1/z)−iβ, |z|=1, α> −1/2, ∈ R. Here C is the normalisation constant chosen so that w0=12π. We use the same approach to compute a difference equation for expressions related to the determinants of the symbol w(z)=et (z+1/z), a symbol important in the study of random permutations. Finally, we study the analogous equations for the symbol w(z)=etz MПα=1 (z-aα/z), gα.

In this paper we show how polynomial mappings of degree K from a union of disjoint intervals onto [−1, 1] generate a countable number of special cases of generalizations of Chebyshev polynomials. We also derive a new expression for these generalized Chebyshev polynomials for any genus g, from which the coefficients of xn can be found explicitly in terms of the branch points and the recurrence coefficients. We find that this representation is useful for specializing to polynomial mapping cases for small K where we will have explicit expressions for the recurrence coefficients in terms of the branch points. We study in detail certain special cases of the polynomials for small degree mappings and prove a theorem concerning the location of the zeroes of the polynomials. We also derive an explicit expression for the discriminant for the genus 1 case of our Chebyshev polynomials that is valid for any configuration of the branch point.

We obtain the rate of decay of the smallest eigenvalue of the Hankel matrices (∫tj+kW2(t) dt)n j,k=0 for a general class of even exponential weights W2=exp(−2Q) on an interval I. More precise asymptotics for more special weights have been obtained by many authors.