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This is a survey paper on an old topic in classical analysis. We present some new developments in asymptotics in the last 50 years. We start with the classical method of Darboux and its generalizations, including a uniformity treatment which has a direct application to the Heisenberg polynomials. We then present the development of an asymptotic theory for difference equations, which is a major advancement since the work of Birkhoff and Trjitzinsky in 1933. A new method was introduced into this field in the 1990s, which is now known as the nonlinear steepest descent method or the Riemann–Hilbert approach. The advantage of this method is that it can be applied to orthogonal polynomials which do not satisfy any differential or difference equations neither do they have any integral representations. As an example, we mention the case of orthogonal polynomials with respect to the Freud weight. Finally, we show how the Wiener–Hopf technique can be used to derive asymptotic expansions for the solutions of an integral equation on a half line. [ABSTRACT FROM AUTHOR]

We consider the asymptotics of the partition function of the extended Gross–Witten–Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with entries expressed in terms of the modified Bessel functions of the first kind and furnishes a τ$\tau$‐function sequence of the Painlevé III′$\text{III}^{\prime }$ equation. We derive the asymptotic expansions of the Toeplitz determinant up to and including the constant terms as the size of the determinant tends to infinity. The constant terms therein are expressed in terms of the Riemann zeta‐function and the Barnes G$G$‐function. A third‐order phase transition in the leading terms of the asymptotic expansions is also observed. [ABSTRACT FROM AUTHOR]

We consider the circular unitary ensemble with a Fisher–Hartwig singularity of both jump type and root type at z = 1 . A rescaling of the ensemble at the Fisher–Hartwig singularity leads to the confluent hypergeometric kernel. By studying the asymptotics of the Toeplitz determinants, we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the Hamiltonian of the coupled Painlevé V system in dimension four. This leads to a Painlevé-type representation of the confluent hypergeometric-kernel determinant. Moreover, the large gap asymptotics, including the constant terms, are derived by evaluating the total integral of the Hamiltonian. In particular, we reproduce the large gap asymptotics of the confluent hypergeometric-kernel determinant obtained by Deift, Krasovsky and Vasilevska, and the sine-kernel determinant as a special case, including the constant term conjectured earlier by Dyson. [ABSTRACT FROM AUTHOR]

We apply the uniform asymptotics method proposed by Bassom et al. to a special Painlevé V equation, and we provide a simpler and more rigorous proof of the connection formulas for a special solution of the equation, which have been established earlier by McCoy and Tang via the isomonodromy and WKB methods. [ABSTRACT FROM AUTHOR]

The Clarkson-McLeod solutions of the fourth Painlevé equation behave like κ D α − 1 2 2 (2 x) as x → + ∞ , where κ is some real constant and D α − 1 2 (x) is the parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we derive the asymptotic behaviors for this class of solutions as x → − ∞. This completes a proof of Clarkson and McLeod's conjecture on the asymptotics of this family of solutions. The total integrals of the Clarkson-McLeod solutions and the asymptotic approximations of the σ -form of this family of solutions are also derived. Furthermore, we find a determinantal representation of the σ -form of the Clarkson-McLeod solutions via an integrable operator with the parabolic cylinder kernel. [ABSTRACT FROM AUTHOR]

Aim: To examine the change in Body Mass Index (BMI) and waist circumference (WC) among primary and secondary school students aged 7-18 in Anhui Province between 2005-2010. Subjects and methods: A total of 15 812 primary and secondary school students aged 7-18 were included in two national surveys on students' constitution and health in 2005 and 2010 in Anhui Province. Measurements of height, weight and WC were taken by trained investigators. BMI was calculated for each subject. Results: The mean BMI and WC were significantly increased from 2005 to 2010. For boys, mean increases were 0.85 kg/m2 and 2.01 cm or 0.08 and 0.34 SD score units, while for girls those were 0.39 kg/m2 and 2.10 cm or 0.20 and 0.39 SD score units (all p < 0.001). The standard deviation scores for WC increased much more than for BMI (0.34 units vs 0.08 units for boys and 0.39 units vs 0.20 units for girls). Conclusions: The average value of BMI and WC among primary and secondary school students has increased dramatically from 2005 to 2010 in Anhui, China. Changes in WC have greatly exceeded those in BMI, showing that WC is a good proxy for central fatness rather than BMI. [ABSTRACT FROM AUTHOR]

We demonstrate an approach to derive uniform and point-wise asymptotic formulas based on Fokas' transform method. To this aim, we study the high-frequency uniform asymptotics for the solution of the Helmholtz equation, in a quarter plane and subject to a specific Neumann condition. The analysis is based on the integral representation of the solution derived via Fokas' transform method. In the case of piecewise constant boundary data, full point-wise asymptotic expansions of the solution are obtained by using the method of steepest descents for definite integrals. A uniform asymptotic expansion, holding in the whole quadrant, is also derived in terms of the complementary error function. The uniform expansion exhibits smooth transitions across certain critical vertical lines, along which the point-wise asymptotic approximations have jumps. [ABSTRACT FROM AUTHOR]

We study the Jacobi unitary ensemble defined by the perturbed Chebyshev weight w(x)=(1-x2)-12(tn2-x2)α, x ∈ (-1, 1), tn > 1. The algebraic singularity coalesces with the end point of the support of the weight function, namely, tn → 1, the hard edge, as n → ∞. We use the Riemann-Hilbert approach (the method of Deift and Zhou) to obtain the asymptotics of the polynomials orthogonal with respect to w(x). We show that the universality property is preserved: the asymptotic behavior of the eigenvalue correlations in the bulk of the spectrum is described in terms of the sine kernel. The main result is on the local behavior at the edge of the spectrum. The limit kernel at the edge, when tn - 1 = O(n-2), is described by the ψ -function for a specific solution of the Painlevé V equation. We also show that when tn varies to a fixed t > 1, the Painlevé V kernel transits to the Bessel kernel J-12. On the other hand, when tn approaches 1, the Painlevé V kernel can be approximated by another Bessel kernel Jα-12. [ABSTRACT FROM AUTHOR]

We study the high-frequency asymptotics for the solution of the modified Helmholtz equation, in a quarter plane and subject to a specific Neumann condition, as an illustration of the procedure to extract the asymptotics via Fokas' transform method. The approach is based on Fokas' integral representation of the solution. Full asymptotic expansions of the solution are obtained by using Watson's lemma and the method of steepest descents for definite integrals. [ABSTRACT FROM PUBLISHER]

Abstract: We study the asymptotic distribution of the zeros of certain classes of Gauss hypergeometric polynomials. Classical analytic methods such as Watson’s lemma and the method of steepest descents are used to analyze the large-n behavior of the zeros of the Gauss hypergeometric polynomials:where n is a nonnegative integer, the constants k, a, τ, λ >0, p >1, and . Owing to the connection between Jacobi polynomials and Gauss hypergeometric polynomials, we prove a special case of a conjecture made by Martínez–Finkelshtein, Martínez–González, and Orive. Numerical evidence and graphical illustrations of the clustering of zeros on certain curves are generated by Mathematica (Version 4.0). [Copyright &y& Elsevier]

We study the asymptotic behavior of the eigenvalue correlations for the unitary ensemble associated with a family of singular weights w(x; μ) = exp { - (1 - x2)-μ}, x ∈ ( - 1, 1) for μ > 0. When μ ∈ (0, 1/2) these are Szegö class weights, and are non-Szegö when μ >= 1/2. It is proved that the behavior in the bulk of the spectrum is described in terms of the sine kernel, which persists the so-called universality results. While the local behavior at the edge of the spectrum is described in terms of the Airy kernel. A specific scaling of the limit reflects the singular behavior of orthogonal polynomials on [ - 1, 1], with respect to the weight w(x; μ). [ABSTRACT FROM AUTHOR]

Abstract: We study a class of sieved Pollaczek polynomials defined by a second-order difference equation (three-term recurrence relation). The measure of orthogonality is determined by using the Markov theorem and the Perron–Stieltjes inversion formula, and is shown consisting of an absolutely continuous part and a discrete part with infinitely many mass points. Uniform asymptotic approximations of these polynomials for large degree n are derived at a turning point and a critical point , involving respectively the Airy function Ai, and . Darboux''s method, the method of steepest descents, and various uniform asymptotic techniques such as cubic transformations are used to derive the results. Asymptotic formulas for the least zeros, the largest zeros, and the zeros on both sides of are also obtained. Several numerical examples are provided to compare the approximate zeros with the true values. [Copyright &y& Elsevier]

Abstract: Uniform asymptotic properties of the classical Jacobi polynomials have been studied via various approaches other than Darboux''s method. In this note, by using ideas of the uniform treatment of Darboux''s method, an asymptotic expansion, in terms of the Bessel function of the first kind and its derivative, is obtained for Jacobi polynomials . The expansion is uniformly valid for , being an arbitrary positive constant. [Copyright &y& Elsevier]

Abstract: Two uniform asymptotic expansions are obtained for the Pollaczek polynomials . One is for , , in terms of elementary functions and in descending powers of . The other is for , in terms of a special function closely related to the modified parabolic cylinder functions, in descending powers of n. This interval contains a turning point and all possible zeros of in . [Copyright &y& Elsevier]

Let F(z) be an analytic function in |z|<1. If F(z) has only a finite number of algebraic singularities on the unit circle |z| = 1, then Darboux’s method can be used to give an asymptotic expansion for the coefficient of zn in the Maclaurin expansion of F(z). However, the validity of this expansion ceases to hold, when the singularities are allowed to approach each other. A special case of this confluence was studied by Fields in 1968. His results have been considered by others to be too complicated, and desires have been expressed to investigate whether any simplification is feasible. In this paper, we shall show that simplification is indeed possible. In the case of two coalescing algebraic singularities, our expansion involves only two Bessel functions of the first kind. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the uniform asymptotic behavior of the single variable Bell polynomials on the negative real axis, to which all zeros belong. It is found that there exists an ascending sequence {Zk}1∞⊂(−e,0) such that the polynomials are represented by a finite sum of infinite asymptotic series, each in terms of the Airy function and its derivative, and the number of series under this sum is 1 in the interval (−∞,Z1) and k+1 in [Zk,Zk+1), k⩾1. Furthermore, it is shown that an asymptotic expansion, also in terms of Airy function and its derivative, completed with error bounds, holds uniformly in (−∞,−δ] for positive δ. [Copyright &y& Elsevier]

The Stokes lines/curves are identified for the Mittag-Leffler function E[sub α,β](z.) = ∑ z[sup n]/Γ(αn + β), Re α > 0. n=0 When α is not real, it is found that the Stokes curves are spirals. Away from the Stokes lines/curves, exponentially improved uniform asymptotic expansions are obtained. Near the Stokes lines/curves, Berry-type smooth transitions are achieved via the use of the complementary error function. [ABSTRACT FROM AUTHOR]

We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we study the generating function of this process on any given number of intervals. It can be expressed as the Fredholm determinant of the confluent hypergeometric kernel with n discontinuities. In this paper, we derive an integral representation for the determinant by using the Hamiltonian of the coupled Painlevé V system. By evaluating the total integral of the Hamiltonian, we obtain the asymptotics of the determinant as the n discontinuities tend to infinity up to and including the constant term. Here the constant term is expressed in terms of the Barnes G -function. • An integral representation for Fredholm determinant of the CHF kernel is obtained. • The Fredholm determinant is related to the Hamiltonian of a coupled Painlevé V system. • The aymptotics of the determinant as discontinuities tend to infinity is derived. [ABSTRACT FROM AUTHOR]

We derive uniform and non-uniform asymptotics of the Charlier polynomials by using difference equation methods alone. The Charlier polynomials are special in that they do not fit into the framework of the turning point theory, despite the fact that they are crucial in the Askey scheme. In this paper, asymptotic approximations are obtained, respectively, in the outside region, an intermediate region, and near the turning points. In particular, we obtain uniform asymptotic approximation at a pair of coalescing turning points with the aid of a local transformation. We also give a uniform approximation at the origin by applying the method of dominant balance and several matching techniques. [ABSTRACT FROM AUTHOR]

In this paper, we consider the monodromy and, in particular, the isomonodromy sets of accessory parameters for the Heun class equations. We show that the Heun class equations can be obtained as limits of the linear systems associated with the Painlevé equations when the Painlevé transcendents go to one of the actual singular points of the linear systems. The isomonodromy sets of accessory parameters for the Heun class equations are described by Taylor or Laurent coefficients of the corresponding Painlevé functions, or the associated tau functions, at the positions of the critical values. As an application of these results, we derive some asymptotic approximations for the isomonodromy sets of accessory parameters in the Heun class equations, including the confluent Heun equation, the doubly‐confluent Heun equation, and the reduced biconfluent Heun equation. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Gaussian unitary ensemble perturbed by a Fisher–Hartwig singularity simultaneously of both root type and jump type. In the critical regime where the singularity approaches the soft edge, namely, the edge of the support of the equilibrium measure for the Gaussian weight, the asymptotics of the Hankel determinant and the recurrence coefficients, for the orthogonal polynomials associated with the perturbed Gaussian weight, are obtained and expressed in terms of a family of smooth solutions to the Painlevé XXXIV equation and the σ‐form of the Painlevé II equation. In addition, we further obtain the double scaling limit of the distribution of the largest eigenvalue in a thinning procedure of the conditioning Gaussian unitary ensemble, and the double scaling limit of the correlation kernel for the critical perturbed Gaussian unitary ensemble. The asymptotic properties of the Painlevé XXXIV functions and the σ‐form of the Painlevé II equation are also studied. [ABSTRACT FROM AUTHOR]

In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight e − n ( z − log z + t / z ) for t < 0 and z on certain contours in the complex plane. When the parameters n , t and the degree k are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy τ -function of the Painlevé III equation. When the degree k = n , n is large and t is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients and the Hankel determinant are described in terms of a Boutroux tronquée solution to the Painlevé I equation. Our approach is based on the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems. [ABSTRACT FROM AUTHOR]

We study the asymptotic expansion for the Landau constants G n , π G n ∼ ln ( 16 N ) + γ + ∑ k = 1 ∞ α k N k as n → ∞ , where N = n + 1 , and γ is Euler’s constant. We show that the signs of the coefficients α k demonstrate a periodic behavior such that ( − 1 ) l ( l + 1 ) 2 α l + 1 < 0 for all l . We further prove a conjecture of Granath which states that ( − 1 ) l ( l + 1 ) 2 ε l ( N ) < 0 for l = 0 , 1 , 2 , … and n = 0 , 1 , 2 , … , ε l ( N ) being the error due to truncation at the l th order term. Consequently, we also obtain the sharp bounds up to arbitrary orders of the form ln ( 16 N ) + γ + ∑ k = 1 p α k N k < π G n < ln ( 16 N ) + γ + ∑ k = 1 q α k N k for all n = 0 , 1 , 2 , … , all p = 4 s + 1 , 4 s + 2 and q = 4 m , 4 m + 3 , with s = 0 , 1 , 2 , … and m = 0 , 1 , 2 , … . [ABSTRACT FROM AUTHOR]

We study the Hankel determinants associated with the weight [ABSTRACT FROM AUTHOR]

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight w ( x ) = x α e − x − t / x , x ∈ ( 0 , ∞ ) , t > 0 and α > 0 . When the matrix size n → ∞ , we obtain an asymptotic formula for the Hankel determinants, valid uniformly for t ∈ ( 0 , d ] , d > 0 fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large- n asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift–Zhou nonlinear steepest descent method. [ABSTRACT FROM AUTHOR]

In this paper, we study the singularly perturbed Laguerre unitary ensemble with $${V_t(x) = x + t/x,\,\, x \in (0,+\infty)}$$ and t > 0. Due to the effect of t/ x for varying t, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves $${\psi}$$ -functions associated with a special solution to a new third-order nonlinear differential equation, which is then shown to be equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter t changes in a finite interval (0, d]. Our approach is based on Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems. [ABSTRACT FROM AUTHOR]

We consider the Clarkson–McLeod solutions of the fourth Painlevé equation. This family of solutions behave like κ D α − 1 2 2 (2 x) as x → + ∞ , where κ is an arbitrary real constant and D α − 1 2 (x) is the parabolic cylinder function. Using the Deift–Zhou nonlinear steepest descent method, we obtain the singular asymptotics of the solutions as x → − ∞ when κ κ − κ ∗ > 0 for some real constant κ ∗. The connection formulas are also explicitly evaluated. This proves and extends Clarkson and McLeod's conjecture that when the parameter κ > κ ∗ > 0 , the Clarkson–McLeod solutions have infinitely many simple poles on the negative real axis. • The Clarkson–McLeod solutions for PIV are those decaying exponentially fast at + ∞. • Clarkson–McLeod conjectured that there will be a blow-up on the negative real axis. • Indeed, there are infinitely many blow-ups on the negative real axis in that case. [ABSTRACT FROM AUTHOR]

We study the uniform asymptotics of a system of polynomials orthogonal on [-1, 1] with weight function w(x) = exp{-1/(1 - x2)μ}, 0 < μ < 1/2, via the Riemann-Hilbert approach. These polynomials belong to the Szegö class. In some earlier literature involving Szegö class weights, Bessel-type parametrices at the endpoints ±1 are used to study the uniform large degree asymptotics. Yet in the present investigation, we show that the original endpoints ±1 of the orthogonal interval are to be shifted to the MRS numbers ±βn, depending on the polynomial degree n and serving as turning points. The parametrices at ±βn are constructed in shrinking neighborhoods of size 1 - βn, in terms of the Airy function. The polynomials exhibit a singular behavior as compared with the classical orthogonal polynomials, in aspects such as the location of the extreme zeros, and the approximation away from the orthogonal interval. The singular behavior resembles that of the typical non-Szegö class polynomials, cf. the Pollaczek polynomials. Asymptotic approximations are obtained in overlapping regions which cover the whole complex plane. Several large-n asymptotic formulas for πn(1), i.e. the value of the nth monic polynomial at 1, and for the leading and recurrence coefficients, are also derived. [ABSTRACT FROM AUTHOR]

In this paper, we develop the Riemann-Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an accumulation point. To illustrate our method, we consider the Tricomi-Carlitz polynomials where is a positive parameter. Uniform Plancherel-Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [W. M. Y. Goh and J. Wimp, On the asymptotics of the Tricomi-Carlitz polynomials and their zero distribution. I, SIAM J. Math. Anal. 25 (1994) 420-428] and in [K. F. Lee and R. Wong, Uniform asymptotic expansions of the Tricomi-Carlitz polynomials, Proc. Amer. Math. Soc. 138 (2010) 2513-2519]. [ABSTRACT FROM AUTHOR]

We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollaczek polynomials as an example. The Plancherel–Rotach type asymptotics of the sieved Pollaczek polynomials are obtained in the whole complex plane. The Riemann–Hilbert method is applied to derive the results. A main feature of the treatment is the appearance of a new band consisting of two adjacent intervals, one of which is a portion of the support of the absolutely continuous measure, the other is the discrete band. [ABSTRACT FROM AUTHOR]