moment matrices and multi component kp with applications to random matrix theory questions on random matrices and non intersecting brownian motions have led to the study of moment matrices with regard to several weights the main result of this paper is to show that the determinants of such moment matrices satisfy upon adding one set of time deformations for each weight the multi component kp hierarchy these determinants are thus tau functions for these integrable hierarchies the tau functions so obtained with appropriate shifts of the time parameters forward and backwards will be expressed in terms of multiple orthogonal polynomials for these weights and their cauchy transforms the main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials it concerns an identity between the orthogonality of polynomials on the real line the bilinear identity in kp theory and a generating functional for the full kp theory an additional fact not discussed in this paper is that these functions satisfy virasoro constraints with respect to these time parameters

differential equations for multiple orthogonal polynomials with respect to classical weights raising and lowering operators we obtain a lowering operator for multiple orthogonal polynomials having orthogonality conditions with respect to r n classical weights these multiple orthogonal polynomials are generalizations of the classical orthogonal polynomials combining the lowering operator with the raising operators which have been obtained earlier in the literature we then also obtain a linear differential equation of order r 1

on 2d discrete schrodinger operators associated with multiple orthogonal polynomials a class of cross shaped difference operators on a two dimensional 2d lattice is introduced the main feature of the operators in this class is that their formal eigenvectors consist of multiple orthogonal polynomials in other words this scheme generalizes the classical connection between jacobi matrices and orthogonal polynomials to the case of operators on lattices furthermore we also show how to obtain 2d discrete schrodinger operators out of this construction and give a number of explicit examples based on known families of multiple orthogonal polynomials

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

on the zeros of d symmetric d orthogonal polynomials abstract in this paper we give some properties of the zeros of d symmetric d orthogonal polynomials and we localize these zeros on d 1 rays emanating from the origin we apply the obtained results to some known polynomials in particular we partially solve the conjecture about the zeros of the humbert polynomials stated by milovanovic and dordevic g v milovanovic g b dordevic on some properties of humbert s polynomials ii ser math inform 6 1991 23 30 a study of the eigenvalues of a particular banded hessenberg matrix is done

convergence of the vector qd algorithm zeros of vector orthogonal polynomials abstract the convergence of the vector qd algorithm associated to d meromorphic functions is established as a consequence a de montessus de ballore theorem for vector pade approximants is proved a short numerical study is done in conclusion

the operator moment problem vector continued fractions and an explicit form of the favard theorem for vector orthogonal polynomials abstract we study the operator moment problem for a p 2 diagonal nonsymmetric matrix operator and give an explicit form of the favard theorem for vector orthogonal polynomials as a main tool we use the development of a system of resolvent functions of the operator in a vector continued fraction by a modified jacobi perron algorithm

generalizations of orthogonal polynomials we give a survey of recent generalizations of orthogonal polynomials that includes multidimensional matrix and vector orthogonal polynomials and multivariate versions multipole orthogonal rational functions variants and extensions of the orthogonality conditions multiple orthogonality most of these generalizations are inspired by the applications in which they are applied we also give a glimpse of these applications which are usually generalizations of applications where classical orthogonal polynomials also play a fundamental role moment problems numerical quadrature rational approximation linear algebra recurrence relations and random matrices

laplace approximations for sums of independent random vectors we consider expressions of the form zn e exp nf sn n where sn is the sum of n i i d random vectors with values in a banach space and f is a smooth real valued function by results of donsker varadhan and bahadur zabell one knows that lim 1 n log zn supxf x h x where h is the so called entropy function in an earlier paper a more precise evaluation of zn is given in the case where there was a unique point maximizing f h and the curvature at the maximum was nonvanishing the present paper treats the more delicate problem where these conditions fail to hold

d orthogonality of little q laguerre type polynomials abstract in this paper we solve a characterization problem in the context of the d orthogonality that allows us on one hand to provide a q analog for the d orthogonal polynomials of laguerre type introduced by the first author and douak and on the other hand to derive new l q classical d orthogonal polynomials generalizing the little q laguerre polynomials various properties of the resulting basic hypergeometric polynomials are singled out for d 1 we obtain a characterization theorem involving as far as we know new l q classical orthogonal polynomials for which we give the recurrence relation and the difference equation

convergence from divergence we show how to convert divergent series which typically occur in many applications in physics into rapidly convergent inverse factorial series this can be interpreted physically as a novel resummation of perturbative series being convergent these new series allow rigorous extrapolation from an asymptotic region with a large parameter to the opposite region where the parameter is small we illustrate the method with various physical examples and discuss how these convergent series relate to standard methods such as borel summation and also how they incorporate the physical stokes phenomenon we comment on the relation of these results to dyson s physical argument for the divergence of perturbation theory this approach also leads naturally to a wide class of relations between bosonic and fermionic partition functions and klein gordon and dirac determinants

on optimal truncation of divergent series solutions of nonlinear differential systems we prove that for divergent series solutions of nonlinear or linear differential systems near a generic irregular singularity the common prescription of summation to the least term is if properly interpreted meaningful and correct and we extend this method to trans series solutions in every direction in the complex plane at the singularity stokes9 directions not excepted there exists a non empty set of solutions whose difference from the optimally i e near the least term truncated asymptotic series is of the same exponentially small order of magnitude as the least term of the series there is a family of generalized borel summation formulae b which commute with the usual algebraic and analytic operations addition multiplication differentiation etc we show that there is exactly one of them b 0 such that for any formal series solution f b 0 f differs from the optimal truncation of f by at most the order of the least term of f we show in addition that the berry smoothing phenomenon is universal within this class of differential systems whenever the terms beyond all orders change in crossing a stokes line these terms vary smoothly on the berry scale x x 1 2 and the transition is always given by the error function under the same conditions we show that dingle9 rule of signs for stokes9 transitions holds

the sherrington kirkpatrick model of spin glasses and stochastic calculus the high temperature case we study the fluctuations of free energy energy and entropy in the high temperature regime for the sherrington kirkpatrick model of spin glasses we introduce here a new dynamical method with the help of brownian motions and continuous martingales indexed by the square root of the inverse temperature as parameter thus formulating the thermodynamic formalism in terms of random processes the well established technique of stochastic calculus leads us naturally to prove that these fluctuations are simple gaussian processes with independent increments a generalization of a result proved by aizenman lebowitz and ruelle 1

on a system of classical polynomials of simultaneous orthogonality abstract we introduce a system of classical polynomials of simultaneous orthogonality study the differential equation of third order recurrence relation and precise the ratio asymptotic and zeros distribution of polynomials

fluctuations of the free energy in the rem and the p spin sk models we consider the random fluctuations of the free energy in the p spin version of the sherrington kirkpatrick sk model in the high temperature regime using the martingale approach of comets and neveu as used in the standard sk model combined with truncation techniques inspired by a recent paper by talagrand on the p spin version we prove that the random corrections to the free energy are on a scale n p 2 2 only and after proper rescaling converge to a standard gaussian random variable this is shown to hold for all values of the inverse temperature smaller than a critical p we also show that p 2 in 2 as p additionally we study the formal p limit of these models the random energy model here we compute the precise limit theorem for the properly rescaled partition function at all temperatures for 2 in 2 fluctuations are found at an exponentially small scale with two distinct limit laws above and below a second critical value in 2 2 for up to that value the rescaled fluctuations are gaussian while below that there are non gaussian fluctuations driven by the poisson process of the extreme values of the random energies for larger than the critical 2 in 2 the fluctuations of the logarithm of the partition function are on a scale of 1 and are expressed in terms of the poisson process of extremes at the critical temperature the partition function divided by its expectation converges to 1 2

some discrete multiple orthogonal polynomials in this paper we extend the theory of discrete orthogonal polynomials on a linear lattice to polynomials satisfying orthogonality conditions with respect to r positive discrete measures first we recall the known results of the classical orthogonal polynomials of charlier meixner kravchuk and hahn t s chihara an introduction to orthogonal polynomials gordon and breach new york 1978 r koekoek and r f swarttouw reports of the faculty of technical mathematics and informatics no 98 17 delft 1998 a f nikiforov et al classical orthogonal polynomials of a discrete variable springer berlin 1991 these polynomials have a lowering and raising operator which give rise to a rodrigues formula a second order difference equation and an explicit expression from which the coefficients of the three term recurrence relation can be obtained then we consider r positive discrete measures and define two types of multiple orthogonal polynomials the continuous case jacobi laguerre hermite etc was studied by van assche and coussement j comput appl math 127 2001 317 347 and aptekarev et al multiple orthogonal polynomials for classical weights manuscript the families of multiple orthogonal polynomials of type ii that we will study have a raising operator and hence a rodrigues formula this will give us an explicit formula for the polynomials finally there also exists a recurrence relation of order r 1 for these multiple orthogonal polynomials of type ii we compute the coefficients of the recurrence relation explicitly when r 2

multiple orthogonal polynomials results on multiple orthogonal polynomials will be surveyed multiple orthogonal polynomials are intimately related to hermite pade approximants and often they are also called hermite pade polynomials special attention will be paid to an application of multiple orthogonal polynomials and to analytic theory of two model families of general multiple orthogonal polynomials referred to as angelesco and nikishin systems among the applications the number theory special functions and spectral analysis of nonsymmetric band operators will be highlighted in the analytic theory results and methods for the study of multiple orthogonal polynomials asymptotics will be reviewed new results on strong asymptotics of multiple orthogonal polynomials for nikishin system will be presented

a riemann hilbert problem for biorthogonal polynomials we characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a riemann hilbert problem our riemann hilbert problem is different from the ones that were proposed recently by ercolani and mclaughlin kapaev and bertola et al we believe that our formulation may be tractable to asymptotic analysis

equilibrium of vector potentials and uniformization of the algebraic curves of genus 0 we consider equilibrium problems for the logarithmic vector potential related to the asymptotics of the hermite pade approximants solutions of such problems can be expressed by means of algebraic functions the goal of this paper is to describe a procedure for determining the algebraic equation for this function in the case when the genus of this algebraic function is equal zero using the coefficients of the equation we compute the extremal cuts of the riemann surfaces these cuts are attractive sets for the poles of the hermite pade approximants we demonstrate the method by an example of the equilibrium problem related to a special system that is called the angelesco system

waves near stokes lines the large k asymptotics of d 2 u z d z 2 k 2 r 2 z u z are studied near a stokes line z z 0 r d z real where z 0 is a zero of r 2 z of any order on which there is greatest disparity between the dominant and subdominant exponential waves in the phase integral wkb approximations the aim is to establish precisely how the multiplier b of the subdominant wave varies across the stokes line although b always has a total change proportional to i times the multiplier of the dominant wave the stokes phenomenon the form of the change depends on the convention used to define the two waves the optimal convention for which the variation is maximally compact and smooth is to define them by the phase integral approximation truncated at its least term whose order is proportional to k and therefore large asymptotics of asymptotics then the variation of b is proportional to the error function of the natural stokes crossing variable im k re this result is obtained without resumming divergent series thereby avoiding asymptotics of asymptotics of asymptotics an application is given to the birth of exponentially weak reflected waves in media with smoothly varying refractive index

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

the sherrington kirkpatrick model a challenge for mathematicians the sherrington kirkpatrick sk model for spin glasses is deceptively simple to state yet its rigorous study represents a considerable challenge we report here some modest progresses obtained through elementary methods even in the supposedly simple high temperature region a number of basic questions remain unsolved

vector orthogonal relations vector qd algorithm vector pade approximants to a function f f 1 f d from to d have been defined uniquely without any auxiliary choice than the degrees of the numerator and the denominator the same for all the components f i as in the scalar case 1 5 the denominators are associated to polynomials p r s which are given by vector orthogonal properties r and which satisfy for each s recurrence relations of order d 1 i e with d 2 terms called relations d we study here consequences of r and d first we prove an algorithm similar to the generalized mna al gorithm then we define a vector qd algorithm which links two diagonals p r s r and p r s 1 r conversely if a family p r s r 0 verifying d is given it is possible to build p r s r 0 s 0 and d linear functionals c a a 1 d such that p r 0 p r and p r s verify the orthogonal relations r with respect to the c a

high temperature regime for a multidimensional sherrington kirkpatrick model of spin glass comets and neveu have initiated in 5 a method to prove convergence of the partition function of disordered systems to a log normal random variable in the high temperature regime by means of stochastic calculus we generalize their approach to a multidimensional sherrington kirkpatrick model with an application to the heisenberg model of uniform spins on a sphere of d see 9 the main tool that we use is a truncation of the partition function outside a small neighbourhood of the typical energy path

the relation of the d orthogonal polynomials to the appell polynomials abstract we are dealing with the concept of d dimensional orthogonal abbreviated d orthogonal polynomials that is to say polynomials verifying one standard recurrence relation of order d 1 among the d orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials in particular we are concerned in the present paper with the solution of the following problem p find all polynomial sequences which are at the same time appell polynomials and d orthogonal the resulting polynomials are a natural extension of the hermite polynomials a sequence of these polynomials is obtained all the elements of its d 1 order recurrence are explicitly determined a generating function a d 1 order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given among such polynomials one singles out the d symmetrical ones definition 1 7 which are the d orthogonal polynomials analogous to the hermite classical ones when d 1 ordinary orthogonality we meet again the classical orthogonal polynomials of hermite

large deviations for langevin spin glass dynamics we study the asymptotic behaviour of asymmetrical spin glass dynamics in a sherrington kirkpatrick model as proposed by sompolinsky zippelius we prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime we study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measureq which is not markovian we deduce that the quenched law of the empirical measure converges to q extending then the preceeding results to replicated dynamics we investigate the quenched behavior of a single spin we get quenched convergence toq in the case of a symmetric initial law and even potential for the free spin

complex jacobi matrices complex jacobi matrices play an important role in the study of asymptotics and zero distribution of formal orthogonal polynomials fops the latter are essential tools in several fields of numerical analysis for instance in the context of iterative methods for solving large systems of linear equations or in the study of pade approximation and jacobi continued fractions in this paper we present some known and some new results on fops in terms of spectral properties of the underlying infinite jacobi matrix with a special emphasis to unbounded recurrence coefficients here we recover several classical results for real jacobi matrices the inverse problem of characterizing properties of the jacobi operator in terms of fops and other solutions of a given three term recurrence is also investigated this enables us to give results on the approximation of the resolvent by inverses of finite sections with applications to the convergence of pade approximants

d orthogonality via generating functions in this paper starting from a suitable generating function of a polynomial set we show how to decide whether the considered polynomial set is d orthogonal and if it is so how to determine the corresponding d dimensional functional vector then we apply the obtained results to some known and new d orthogonal polynomial sets for the known ones we give new proofs for some already obtained results

on semiclassical linear functionals integral representations abstract if is a functional of moments and satisfies the equation d o where x and x are arbitrary polynomials but with the only condition deg 1 then is said to be a semiclassical functional when is semiclassical and regular its corresponding orthogonal polynomial sequence ops pn x n 0 is called a semiclassical ops in this paper integral representations are given for semiclassical functionals regular or not in cases a deg deg and b deg deg but in the latter case when deg 0 the fundamental result is the following if n with n 0 are the moments of an a functional then n cm n this allows us to find integral representations for a functionals using standard techniques taken from pollaczek 1951