Alfredo Deaño

Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature

In this paper we study the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. The zeros of these polynomials are the nodes for complex Gaussian quadrature of an oscillatory integral on the real axis with a high order stationary point, and their limit distribution is also analyzed. We show that the zeros accumulate along a contour in the complex plane that has the S-property in an external field. In addition, the strong asymptotics of the orthogonal polynomials are obtained by applying the nonlinear Deift-Zhou steepest descent method to the corresponding Riemann-Hilbert problem.

On second-order differential equations with highly oscillatory forcing terms

We present a method to compute efficiently solutions of systems of ordinary differential equations (ODEs) that possess highly oscillatory forcing terms. This approach is based on asymptotic expansions in inverse powers of the oscillatory parameter, and features two fundamental advantages with respect to standard numerical ODE solvers: first, the construction of the numerical solution is more efficient when the system is highly oscillatory, and, second, the cost of the computation is essentially independent of the oscillatory parameter. Numerical examples are provided, featuring the Van der Pol and Duffing oscillators and motivated by problems in electronic engineering.

New inequalities from classical Sturm theorems

Inequalities satisfied by the zeros of the solutions of second-order hypergeometric equations are derived through a systematic use of Liouville transformations together with the application of classical Sturm theorems. This systematic study allows us to improve previously known inequalities and to extend their range of validity as well as to discover inequalities which appear to be new. Among other properties obtained, Szegos bounds on the zeros of Jacobi polynomials Pn(α,β) (cos θ) for |α| < ½, |β| < ½ are completed with results for the rest of parameter values, Grosjeans inequality (J. Approx. Theory 50 (1987)84) on the zeros of Legendre polynomials is shown to be valid for Jacobi polynomials with |β| ≤ 1, bounds on ratios of consecutive zeros of Gauss and confluent hypergeometric functions are derived as well as an inequality involving the geometric mean of zeros of Bessel functions.

Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions

Three term recurrence relations yn+1 +bnyn +anyn?1 = 0 can be used for computing recursively a great number of special functions. Depending on the asymptotic nature of the function to be computed, different recursion directions need to be considered: backward for minimal solutions and forward for dominant solutions. However, some solutions interchange their role for finite values of n with respect to their asymptotic behaviour and certain dominant solutions may transitorily behave as minimal. This phenomenon, related to Gautschi’s anomalous convergence of the continued fraction for ratios of confluent hypergeometric functions, is shown to be a general situation which takes place for recurrences with an negative and bn changing sign once. We analyze the anomalous convergence of the associated continued fractions for a number of different recurrence relations (modified Bessel functions, confluent and Gauss hypergeometric functions) and discuss the implication of such transitory behaviour on the numerical stability of recursion.

On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials

In this paper, we compute the probability that an N x N matrix from the generalized Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar (2008). For this purpose, we work out the large degree asymptotics of semi-classical Laguerre polynomials and their recurrence coefficients, using the steepest descent analysis of the corresponding Riemann–Hilbert problem.

Painlevé I double scaling limit in the cubic random matrix model

We obtain the double scaling asymptotic behavior of the recurrence coefficients and the partition function at the critical point of the N × N Hermitian random matrix model with cubic potential. We prove that the recurrence coefficients admit an asymptotic expansion in powers of N−2/5, and in the leading order the asymptotic behavior of the recurrence coefficients is given by a Boutroux tronquee solution to the Painleve I equation. We also obtain the double scaling limit of the partition function, and we prove that the poles of the tronquee solution are limits of zeros of the partition function. The tools used include the Riemann–Hilbert approach and the Deift–Zhou nonlinear steepest descent method for the corresponding family of complex orthogonal polynomials and their recurrence coefficients, together with the Toda equation in the parameter space.

Asymptotic solvers for oscillatory systems of differential equations

We describe an asymptotic method for approximating solutions of systems of ODEs with oscillatory forcing terms. The approach is based on asymptotic expansions in inverse powers of the oscillatory parameter ω and on modulated Fourier expansions. We revise some relevant examples, including problems that appear in the modelling of mechanical and electronic systems, and for which the proposed procedure is superior to standard methods.

Construction and implementation of asymptotic expansions for Jacobi---type orthogonal polynomials

AbstractWe are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to ∞

Analytical and Numerical Aspects of a Generalization of the Complementary Error Function

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Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity

. These are defined on the interval [−1, 1] with weight function w(x)=(1−x)α(1+x)βh(x),α,β>−1