Portrait of Dr Alfredo Deaño

Dr Alfredo Deaño

Lecturer in Mathematics


Alfredo joined SMSAS in 2015 from Universidad Carlos III de Madrid (Spain), where he worked as a postdoctoral assistant in the Department of Mathematics. He completed two postdoctoral research stays in the Department of Computer Science at KU Leuven (2012-2014) and in the Department of Applied Mathematics and Theoretical Physics (DAMTP) at University of Cambridge (2007-2009). He obtained his PhD from Universidad Carlos III de Madrid in 2006, and a BSc in Mathematics from Universidad Autónoma de Madrid (Spain) in 2001. Additionally he completed the degree in music (flute) in Madrid in 1999.

Research interests

  • Special functions and orthogonal polynomials
  • Painlevé equations 
  • Random matrices 
  • Asymptotic and numerical analysis 


Currently principal investigator (PI) of the EPSRC project Painlevé equations: analytical properties and numerical computation



  • Charlier, C. and DeañoA. (2018). Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity. Symmetry, Integrability and Geometry: Methods and Applications [Online] 14. Available at: https://doi.org/10.3842/SIGMA.2018.018.
    We study n × n Hankel determinants constructed with moments of a Hermite
    weight with a Fisher–Hartwig singularity on the real line. We consider the case when
    the singularity is in the bulk and is both of root-type and jump-type. We obtain large n
    asymptotics for these Hankel determinants, and we observe a critical transition when the
    size of the jumps varies with n. These determinants arise in the thinning of the generalised
    Gaussian unitary ensembles and in the construction of special function solutions of the
    Painleve IV equation.
  • DeañoA. (2018). Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane. Symmetry, Integrability and Geometry: Methods and Applications [Online] 14. Available at: https://doi.org/10.3842/SIGMA.2018.107.
    In this paper we obtain large z asymptotic expansions in the complex plane for
    the tau function corresponding to special function solutions of the Painlev´e II differential
    equation. Using the fact that these tau functions can be written as n × n Wronskian
    determinants involving classical Airy functions, we use Heine’s formula to rewrite them as
    n-fold integrals, which can be asymptotically approximated using the classical method of
    steepest descent in the complex plane.
  • DeañoA. and SImm, N. (2017). On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials. Journal of Approximation Theory [Online] 220:44-59. Available at: https://doi.org/10.1016/j.jat.2017.04.004.
    In this paper, we compute the probability that an NxN matrix from the generalised Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar. 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.
  • DeañoA., Huertas, E. and Román, P. (2016). Asymptotics of orthogonal polynomials generated by a Geronimus perturbation of the Laguerre measure. Journal of Mathematical Analysis and Applications [Online] 433:732-746. Available at: https://doi.org/10.1016/j.jmaa.2015.08.002.
    This paper deals with monic orthogonal polynomials generated by a Geronimus canonical spectral transformation of the Laguerre classical measure for x in [0,?), ? > ?1, a free parameter N and a shift c<0. We analyze the asymptotic behavior (both strong and relative to classical Laguerre polynomials) of these orthogonal polynomials as n tends to infinity.
  • DeañoA., Huybrechs, D. and Opsomer, P. (2016). Construction and implementation of asymptotic expansions for Jacobi-type orthogonal polynomials. Advances in Computational Mathematics [Online]:1-32. Available at: http://www.dx.doi.org/10.1007/s10444-015-9442-z.
    We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to infinity. These are defined on the interval [?1,1] with weight function w(x)=(1?x)^a(1+x)^b h(x), a,b>?1 and h(x) a real, analytic and strictly positive function on [?1,1]. This information is available in the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen, where the authors use the Riemann--Hilbert formulation and the Deift-Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in O(n) operations, rather than O(n^2) based on the recurrence relation.
  • Bleher, P., DeañoA. and Yattselev, M. (2016). Topological expansions in the complex cubic log-gas model. One-cut case. Journal of Statistical Physics [Online] 166:784-827. Available at: http://dx.doi.org/10.1007/s10955-016-1621-x.
    We prove the topological expansion for the cubic log-gas partition function, with a complex parameter and defined on an unbounded contour on the complex plane. The complex cubic log-gas model exhibits two phase regions on the complex t-plane, with one cut and two cuts, separated by analytic critical arcs of the two types of phase transition: split of a cut and birth of a cut. The common point of the critical arcs is a tricritical point of the Painleve I type. In the present paper we prove the topological expansion for the partition function in the one-cut phase region. The proof is based on the Riemann-Hilbert approach to semiclassical asymptotic expansions for the associated orthogonal polynomials and the theory of S-curves and quadratic differentials.
  • Bleher, P. and DeañoA. (2016). Painlevé I double scaling limit in the cubic matrix model. Random Matrices: Theory and Applications [Online] 5:1650004. Available at: http://dx.doi.org/10.1142/S2010326316500040.
  • DeañoA., Kuijlaars, A. and Román, P. (2015). Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions. Constructive Approximation [Online]:1-44. Available at: http://www.dx.doi.org/10.1007/s00365-015-9300-8.
    We consider polynomials P_n orthogonal with respect to the weight J_? on [0,?), where J_? is the Bessel function of order ?. Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros are complex and accumulate as n?? near the vertical line Rez=??2. We prove this fact for the case 0???1/2 from strong asymptotic formulas that we derive for the polynomials Pn in the complex plane. Our main tool is the Riemann-Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift-Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for ??1/2.
  • Condon, M., DeañoA., Gao, J. and Iserles, A. (2015). Asymptotic solvers for ordinary differential equations with multiple frequencies. Science China Mathematics [Online] 58:2279-2300. Available at: http://www.dx.doi.org/10.1007/s11425-015-5066-5.
    We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focusing on the case of multiple, non-commensurate frequencies. We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question. Numerical examples illustrate the effectiveness of the method.
  • DeañoA. (2014). Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval. Journal of Approximation Theory [Online] 186:33-63. Available at: http://www.dx.doi.org/10.1016/j.jat.2014.07.004.
    We consider polynomials p^w_n(x) that are orthogonal with respect to the oscillatory weight w(x)=exp(iwx) on [?1,1], where w>0 is a real parameter. A first analysis of p^?_n(x) for large values of w was carried out in connection with complex Gaussian quadrature rules with uniform good properties in w. In this contribution we study the existence, asymptotic behavior and asymptotic distribution of the roots of p^?_n(x) in the complex plane as n tends to infinity. The parameter w grows with n linearly. The tools used are logarithmic potential theory and the S-property, together with the Riemann--Hilbert formulation and the Deift-Zhou steepest descent method.
  • Asheim, A., DeañoA., Huybrechs, D. and Wang, H. (2014). A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Discrete and Continuous Dynamical Systems - A [Online] 34:883-901. Available at: http://dx.doi.org/10.3934/dcds.2014.34.883.
    We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function eiωx on the interval [−1,1]. We show that such a rule attains high asymptotic order, in the sense that the quadrature error quickly decreases as a function of the frequency ω. However, accuracy is maintained for all values of ω and in particular the rule elegantly reduces to the classical Gauss-Legendre rule as ω→0. The construction of such rules is briefly discussed, and though not all orthogonal polynomials exist, it is demonstrated numerically that rules with an even number of points are well defined. We show that these rules are optimal both in terms of asymptotic order as well as in terms of polynomial order.
  • Condon, M., DeañoA., Gao, J. and Iserles, A. (2014). Asymptotic solvers for second-order differential equation systems with multiple frequencies. Calcolo [Online] 51:109-139. Available at: https://doi.org/10.1007/s10092-013-0078-4.
    In this paper, an asymptotic expansion is constructed to solve second-order differential equation systems with highly oscillatory forcing terms involving multiple frequencies. An asymptotic expansion is derived in inverse of powers of the oscillatory parameter and its truncation results in a very effective method of dicretizing the differential equation system in question. Numerical experiments illustrate the effectiveness of the asymptotic method in contrast to the standard Runge-Kutta method
  • DeañoA., Huertas, E. and Marcellán, F. (2013). Strong and ratio asymptotics for Laguerre polynomials revisited. Journal of Mathematical Analysis and Applications [Online] 403:477-486. Available at: https://doi.org/10.1016/j.jmaa.2013.02.039.
    n this paper we consider the strong asymptotic behavior of Laguerre polynomials in the complex plane. The leading behavior is well known from Perron and Mehler–Heine formulas, but higher order coefficients, which are important in the context of Krall–Laguerre or Laguerre–Sobolev-type orthogonal polynomials, are notoriously difficult to compute. In this paper, we propose the use of an alternative expansion, due to Buchholz, in terms of Bessel functions of the first kind. The coefficients in this expansion can be obtained in a straightforward way using symbolic computation. As an application, we derive extra terms in the asymptotic expansion of ratios of Laguerre polynomials in...
  • Bleher, P. and DeañoA. (2012). Topological expansion in the cubic random matrix model. International Mathematics Research Notices [Online] 12:2699-2755. Available at: https://doi.org/10.1093/imrn/rns126.
    In this paper, we study the topological expansion in the cubic random matrix model, and we evaluate explicitly the expansion coefficients for genus 0 and 1. For genus 0 our formula coincides with the one in [6]. For higher genus, we obtain the asymptotic behavior of the coefficients in the expansion as the number of vertices of the associated graphs tends to infinity. Our study is based on the Riemann–Hilbert problem, string equations, and the Toda equation.
  • Condon, M., DeañoA., Iserles, A. and Kropielnicka, K. (2012). Efficient computation of delay differential equations with highly oscillatory terms. ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN) [Online] 46:1407-1420. Available at: https://doi.org/10.1051/m2an/2012004.
    This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation.
  • DeañoA., Huybrechs, D. and Kuijlaars, A. (2010). Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature. Journal of Approximation Theory [Online] 162:2202-2224. Available at: http://dx.doi.org/10.1016/j.jat.2010.07.006.
    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.
  • Condon, M., DeañoA. and Iserles, A. (2010). On systems of differential equations with extrinsic oscillation. Discrete and Continuous Dynamical Systems - A [Online] 28:1345-1367. Available at: http://dx.doi.org/10.3934/dcds.2010.28.1345.
    We present a numerical scheme for an efficient discretization of nonlinear systems of differential equations subject to highly oscillatory perturbations. This method is superior to standard ODE numerical solvers in the presence of high frequency forcing terms, and is based on asymptotic expansions of the solution in inverse powers of the oscillatory parameter ?, featuring modulated Fourier series in the expansion coefficients. Analysis of numerical stability and numerical examples are included.
  • DeañoA. and Temme, N. (2010). Analytical and numerical aspects of a generalization of the complementary error function. Applied Mathematics and Computation [Online] 216:3680-3693. Available at: https://doi.org/10.1016/j.amc.2010.05.025.
  • Condon, M., DeañoA. and Iserles, A. (2010). On second order differential equations with highly oscillatory forcing terms. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences [Online] 466:1809-1828. Available at: http://dx.doi.org/10.1098/rspa.2009.0481.
    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.
  • DeañoA., Segura, J. and Temme, N. (2010). Computational properties of three-term recurrence relations for Kummer functions. Journal of Computational and Applied Mathematics [Online] 233:1505-1510. Available at: http://dx.doi.org/10.1016/j.cam.2008.03.051.
    Several three-term recurrence relations for confluent hypergeometric functions are analyzed from a numerical point of view. Minimal and dominant solutions for complex values of the variable are given, derived from asymptotic estimates of the Whittaker functions with large parameters. The Laguerre polynomials and the regular Coulomb wave functions are studied as particular cases, with numerical examples of their computation.
  • DeañoA. and Huybrechs, D. (2009). Complex Gaussian quadrature of oscillatory integrals. Numerische Mathematik [Online] 112:197-219. Available at: http://dx.doi.org/10.1007/s00211-008-0209-z.
    We construct and analyze Gauss-type quadrature rules with complex- valued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behaviour, suitable for the application of Gaussian rules with non-standard weight functions. The results differ from those in previous research in the sense that the constructed rules are asymptotically optimal, i.e., among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand.
  • Condon, M., DeañoA. and Iserles, A. (2009). On highly oscillatory problems arising in electronic engineering. ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN) [Online] 43:785-804. Available at: https://doi.org/10.1051/m2an/2009024.
    In this paper, we consider linear ordinary differential equations originating in electronic engineering, which exhibit exceedingly rapid oscillation. Moreover, the oscillation model is completely different from the familiar framework of asymptotic analysis of highly oscillatory integrals. Using a Bessel-function identity, we expand the oscillator into asymptotic series, and this allows us to extend Filon-type approach to this setting. The outcome is a time-stepping method that guarantees high accuracy regardless of the rate of oscillation.
  • DeañoA. and Temme, N. (2009). On modified asymptotic series involving confluent hypergeometric functions. Electronic Transactions on Numerical Analysis [Online] 35:88-103. Available at: http://etna.mcs.kent.edu/volumes/2001-2010/vol35/abstract.php?vol=35&pages=88-103.
    A modification of the Poincaré-type asymptotic expansion for functions defined by Laplace transforms is analyzed. This modification is based on an alternative power series expansion of the integrand, and the convergence properties are seen to be superior to those of the original asymptotic series. The resulting modified asymptotic expansion involves a series of confluent hypergeometric functions U(a,c,z), which can be computed by means of continued fractions in a backward recursion scheme. Numerical examples are included, such as the incomplete gamma function ?(a,z) and the modified Bessel function K?(z) for large values of z. It is observed that the same procedure can be applied to uniform asymptotic expansions when extra parameters become large as well.
  • Bueno, M., DeañoA. and Tavernetti, E. (2009). A new algorithm for computing the Geronimus transformations for large shifts. Numerical Algorithms [Online] 54:101-139. Available at: http://dx.doi.org/10.1007/s11075-009-9325-9.
    A monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure. The basic Geronimus transformation with shift ? transforms the monic Jacobi matrix associated with a measure d? into the monic Jacobi matrix associated with d?/(x????)?+?C?(x????), for some constant C. In this paper we examine the algorithms available to compute this transformation and we propose a more accurate algorithm, estimate its forward errors, and prove that it is forward stable. In particular, we show that for C?=?0 the problem is very ill-conditioned, and we present a new algorithm that uses extended precision.
  • DeañoA., Segura, J. and Temme, N. (2008). Identifying minimal and dominant solutions for Kummer recursions. Mathematics of Computation [Online] 77:2277-2293. Available at: https://doi.org/10.1090/S0025-5718-08-02122-4.
  • DeañoA. and Segura, J. (2007). Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions. Mathematics of Computation [Online] 76:879-901. Available at: http://dx.doi.org/10.1090/S0025-5718-07-01934-5.
    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.
  • DeañoA. and Segura, J. (2007). Global Sturm inequalities for the real zeros of the solutions of the Gauss hypergeometric equation. Journal of Approximation Theory [Online] 148:92-110. Available at: https://doi.org/10.1016/j.jat.2007.02.005.
    Liouville–Green transformations of the Gauss hypergeometric equation with changes of variable z(x) = x tp?1(1 ? t)q?1 dt are considered. When p + q = 1, p = 0 or q = 0 these transformations, together with the application of Sturm theorems, lead to properties satisfied by all the real zeros xi of any of its solutions in the interval (0, 1). Global bounds on the differences z(xk+1) ? z(xk), 0 < xk < xk+1 < 1 being consecutive zeros, and monotonicity of these distances as a function of k can be obtained. We investigate the parameter ranges for which these two different Sturm-type properties are available. Classical results for Jacobi polynomials (Szegö’s bounds, Grosjean’s inequality) are particular cases of these more general properties. Similar properties are found for other values of p and q, particularly when |p|=|| and |q|=||, and being the usual Jacobi parameters.
  • DeañoA. and Segura, J. (2004). New inequalities from classical Sturm theorems. Journal of Approximation Theory [Online] 131:208-230. Available at: https://doi.org/10.1016/j.jat.2004.09.006.
    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, Szeg?'s bounds on the zeros of Jacobi polynomials for , are completed with results for the rest of parameter values, Grosjean's inequality (J. Approx. Theory 50 (1987) 84) on the zeros of Legendre polynomials is shown to be valid for Jacobi polynomials with , 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


  • DeañoA., Huybrechs, D. and Iserles, A. (2017). Computing Highly Oscillatory Integrals. Society for Industrial and Applied Mathematics.

Book section

  • DeañoA., Gil, A. and Segura, J. (2006). Computation of real zeros of the Kummer function M(a;c;x). In: Mathematical Software - ICMS 2006 Second International Congress on Mathematical Software. Berlin, Germany: Springer, pp. 296-307. Available at: http://dx.doi.org/10.1007/11832225_30.
    An algorithm for computing the real zeros of the Kummer function M(a;c;x) is presented. The computation of ratios of functions of the type M(a+1; c+1; x)/M(a; c; x), M(a+1; c; x)/M(a; c; x) plays a key role in the algorithm, which is based on global fixed-point iterations. We analyse the accuracy and efficiency of three continued fraction representations converging to these ratios as a function of the parameter values. The condition of the change of variables appearing in the fixed point method is also studied. Comparison with implicit Maple functions is provided, including the Laguerre polynomial case.


  • DeañoA., Huybrechs, D. and Iserles, A. (2015). The kissing polynomials and their Hankel determinants. The kissing polynomials and their Hankel determinants [Online]. Available at: http://arxiv.org/abs/1504.07297.
    We study a family of polynomials that are orthogonal with respect to the weight function exp(iwx) in [?1,1], where w?0. Since this weight function is complex-valued and, for large ?, highly oscillatory, many results in the classical theory of orthogonal polynomials do not apply. In particular, the polynomials need not exist for all values of the parameter ?, and, once they do, their roots lie in the complex plane. Our results are based on analysing the Hankel determinants of these polynomials, reformulated in terms of high-dimensional oscillatory integrals which are amenable to asymptotic analysis. This analysis yields existence of the even-degree polynomials for large values of w, an asymptotic expansion of the polynomials in terms of rescaled Laguerre polynomials near ±1 and a description of the intricate structure of the roots of the Hankel determinants in the complex plane. This work is motivated by the design of efficient quadrature schemes for highly oscillatory integrals.
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