Special non uniform lattice (snulsnul) orthogonal polynomials on discrete dense sets of points.

Difference calculus compatible with polynomials (i.e., such that the divided difference operator of first order applied to any polynomial must yield a polynomial of lower degree) can only be made on special lattices well known in contemporary $q-$calculus. Orthogonal polynomials satisfying difference relations on such lattices are presented. In particular, lattices which are dense on intervals ($|q|=1$) are considered

Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials

AbstractRecurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function w such that w′w is a rational function) are shown to be solutions of nonlinear differential equations with respect to a well-chosen parameter, according to principles established by D. Chudnovsky and G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in an + 1Pn + 1 (x) = xpn(x) − anpn − 1 (x) of the orthogonal polynomials related to the weight exp (− x44 − tx2) on R satisfy 4an3än = (3an4 + 2tan2 − n)(an4 + 2tan2 + n), and an2 satisfies a Painlevé PIV equation

Toeplitz matrix techniques and convergence of complex weight Padé approximants

AbstractOne considers diagonal Padé approximants about ∞ of functions of the form f(z)=∫−11(z−x)−1w(x)dx,z∉[−1,1], where w is an integrable, possibly complex-valued, function defined on [-1, 1].Convergence of the sequence of diagonal Padé approximants towards f is established under the condition that there exists a weight ω, positive almost everywhere on [-1, 1], such that g(x)=w(x)/ω(x) is continuous and not vanishing on [-1, 1].The rate of decrease of the error is also described.The proof proceeds by establishing the link between the Padé denominators and the orthogonal polynomials related to ω, in terms of the Toeplitz matrix of symbol g(cos Φ)

Rational interpolation to solutions of Riccati difference equations on elliptic lattices

AbstractIt is shown how to define difference equations on particular lattices {xn}, n∈Z, where the xns are values of an elliptic function at a sequence of arguments in arithmetic progression (elliptic lattice). Solutions to special difference equations (elliptic Riccati equations) have remarkable simple (!) interpolatory continued fraction expansions