Yurii A. Neretin
We construct noncomplete orthogonal systems on the ray $[0,\infty)$ that look like Jacobi polynomials $P_n(x)$ after a shift of degree $n\mapsto n+a$, where $a$ is a real constant. These systems are solutions of some exotic Sturm-Liouville problem for hypergeometric differential operators. We obtain the explicit spectral decomposition for these problems.
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Hypergeometric Expressions for Type I Jacobi-Piñeiro Orthogonal Polynomials with Arbitrary Number of Weights
本文针对任意数量测度，利用多重超几何函数给出了I型Jacobi-Piñeiro和第一类Laguerre多重正交多项式的显式表达式。
A few identities and integrals involving Pochhammer symbols, Jacobi polynomials, and the generalized hypergeometric function
本文通过Jacobi多项式和广义超几何函数的展开，推导出涉及Pochhammer符号的新的恒等式和积分闭形式。
A problem of determining zeroes of the Gauss hypergeometric function goes back to Klein, Hurwitz, and Van Vleck. In this very short note we show how ratios of hypergeometric functions arise as m-functions of Jacobi matrices and we then revisit the problem based on the recent developments of the spectral theory of non-Hermitian Jacobi matrices.
The algebraic underpinning of the tridiagonalization procedure is investigated. The focus is put on the tridiagonalization of the hypergeometric operator and its associated quadratic Jacobi algebra. It is shown that under tridiagonalization, the quadratic Jacobi algebra becomes the quadratic Racah-Wilson algebra associated to the generic Racah/Wilson polynomials. A degenerate case leading to the Hahn algebra is also discussed.
We study Heun's differential equation in the case that one of the singularities is apparent. In particular we conjecture a relationship with generalized hypergeometric differential equation and establish it in some cases. We apply our results to exceptional Jacobi polynomials.
Matrix-valued analogues of the little q-Jacobi polynomials are introduced and studied. For the 2x2-matrix-valued little q-Jacobi polynomials explicit expressions for the orthogonality relations, Rodrigues formula, three-term recurrence relation and their relation to matrix-valued q-hypergeometric series and the scalar-valued little q-Jacobi polynomials are presented. The study is based on a matrix-valued q-difference operator, which is a q-analogue of Tirao's matrix-valued hypergeometric differential operator.
We first show how one can obtain Al-Salam--Chihara polynomials, continuous dual $q$-Hahn polynomials, and Askey--Wilson polynomials from the little $q$-Laguerre and the little $q$-Jacobi polynomials by using special transformations. This procedure is then extended to obtain multiple Askey--Wilson, multiple continuous dual $q$-Hahn, and multiple Al-Salam--Chihara polynomials from the multiple little $q$-Laguerre and the multiple little $q$-Jacobi polynomials.
We find all polynomials solutions $P_n(x)$ of the abstract "hypergeometric" equation $L P_n(x) = λ_n P_n(x)$, where $L$ is a linear operator sending any polynomial of degree $n$ to a polynomial of the same degree with the property that $L$ is two-diagonal in the monomial basis, i.e. $L x^n = λ_n x^n + μ_n x^{n-1}$ with arbitrary nonzero coefficients $λ_n, μ_n$ . Under obvious nondegenerate conditions, the polynomial eigensolutions $L P_n(x) = λ_n P_n(x)$ are unique. The main result of the paper is a classification of all {\it orthogonal} polynomials $P_n(x)$ of such type, i.e. $P_n(x)$ are ass…
We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hypergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variable totally elliptic hypergeometric series is proved.
We write spectral decomposition of the hypergeometric differential operator on the contour $Re z=1/2$ (multiplicity of spectrum is 2). As a result, we obtain an integral transform that differs from the Jacobi (or Olevsky) transform. We also try to do a step towards vector-valued special functions and construct a 3F2 orthogonal basis in the space of functions having values in 2-dimensional space. This basis is lying in an analytic continuation of continuous dual Hahn polynomials with respect to number $n$ of a polynomial. In Addendum, we discuss a vector-value analog of the Meixner--Pollachek o…
We give an analogy of Jacobi's formula, which relates the hypergeometric function with parameters $(1/4,1/4,1)$ and theta constants. By using this analogy and twice formulas of theta constants, we obtain a transformation formula for this hypergeometric function. As its application, we express the limit of a pair of sequences defined by a mean iteration by this hypergeometric function.
We generalize generating functions for hypergeometric orthogonal polynomials, namely Jacobi, Gegenbauer, Laguerre, and Wilson polynomials. These generalizations of generating functions are accomplished through series rearrangement using connection relations with one free parameter for these orthogonal polynomials. We also use orthogonality relations to determine corresponding definite integrals.
We use a method based on the division algorithm to determine all the values of the real parameters $b$ and $c$ for which the hypergeometric polynomials $_2F_1(-n, b; c; z)$ have $n$ real, simple zeros. Furthermore, we use the quasi-orthogonality of Jacobi polynomials to determine the intervals on the real line where the zeros are located.
We present a method to obtain weight functions associated with linear and quadratic lattices that yield discrete orthogonality with respect to a quasi-definite moment functional of the orthogonal polynomial sequences in the Askey scheme, with the exception of the Jacobi, Bessel, Laguerre, and Hermite polynomials.
In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is studied, and in particular we study solutions given in terms of power series at $0$ and at $\infty$. Factorisations of the corresponding operator are considered in terms of a lowering operator, which is the $q$-derivative, and the related raising operator. Next we consider the $q$-hypergeometric operator in a special case, and we show that there is a natural…
A short introduction to the use of the spectral theorem for self-adjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on l^2(N), leading to a proof of Favard's theorem stating that polynomials satisfying a three-term recurrence relation are orthogonal polynomials. We discuss the link to the moment problem. In the second example, the spectral theorem is applied to Jacobi operators on l^2(Z). We discuss the theorem of Masson and Repka linking the deficiency indices of a Jacobi operato…
The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appell's function $F_{3}(:)$ due to Marichev-Saigo-Maeda (MSM), to the Jacobi type orthogonal polynomials. The results are expressed in terms of generalized hypergeometric function. Some of the interesting special cases of the main results also established.
We prove that the zeros of ${}_2F_1(-n,\frac{n+1}{2};\frac{n+3}{2};z)$ asymptotically approach the section of the lemniscate $\{z: |z(1-z)^2|=4/27; \textrm{Re}(z)>1/3\}$ as $n\rightarrow \infty$. In recent papers (cf. \cite{KMF}, \cite{orive}), Martínez-Finkelshtein and Kuijlaars and their co-authors have used Riemann-Hilbert methods to derive the asymptotic zero distribution of Jacobi polynomials $P_n^{(α_n,β_n)}$ when the limits $\ds A=\lim_{n\rightarrow \infty}\frac{α_n}{n}$ and $\ds B=\lim_{n\rightarrow \infty}\frac{β_n}{n}$ exist and lie in the interior of certain specified regions in the…
We study the perturbative power-series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d. The(small) expansion parameters are being the entries of the two diagonals of length d-1 sandwiching the principal diagonal, which gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series of 3d-5 variables in the generic case, or 2d-3 variables for the eigenvalue growing from a corner matrix element. To derive the result, we first rewrite the spectral problem for a Jacobi matrix as an e…
A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval $(0,1)$ is studied. This type of polynomials have direct applications in the investigation of singular values of products of Ginibre matrices, in the analysis of rational solutions to Painlevé equations and are connected with branched continued fractions and total positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The…
Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the asymptotic representation of the Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Laguerre polynomials.
Szmytkowski derived a certain integral with Gegenbauer polynomials. A natural generalization is to derive lookalike integrals with Jacobi polynomials. Six methods are treated to derive the first integral. The first method should be enough to prove the first integral, but by the other methods there arises remarkable formula such as par example a zero-balanced F3 Appell function which can be converted into a 2F1 hypergeometric function. Another three integrals complete the paper.
Zeros of orthogonal little q-Jacobi polynomials: interlacing and monotonicity
本文利用对数网格研究小 q-Jacobi 多项式零点的交错性与参数单调性，并推广到 q-Bessel 等极限情形，给出非正交区域的结构分解。
By considering Schwarz's map for the hypergeometric differential equation with parameters $(a,b,c)=(1/6,1/2,1)$ or $(1/12,5/12,1)$, we give some analogies of Jacobi's formula $\vartheta_{00}(τ)^2= F(1/2,1/2,1;λ(τ))$, where $\vartheta_{00}(τ)$ and $λ(τ)$ are the theta constant and the lambda function defined on the upper-half plane, and $F(a,b,c;z)$ is the hypergeometric series defined on the unit disk. As applications of our formulas, we give several functional equations for $F(a,b,c;z)$.
In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generating functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given.
For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q=0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relations which admit a dual system of orthogonal polynomials. For the q=0 orthogonal limit functions we discuss interpretations on p-adic spaces. In the little 0-Jacobi case we also discuss product formulas.
We construct the q-twisted cohomology associated with the q-multiplicative function of Jordan-Pochhammer type at |q|=1. In this framework, we prove the Heine's relations and a connection formula for the q-hypergeometric function of the Barnes type. We also prove an orthogonality relation of the q-little Jacobi polynomials at |q|=1.
By applying the Euler--Rayleigh methods to a specific representation of the Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, we derive upper and lower bound for $1-x_{nn}^2(λ)$, with $x_{nn}(λ)$ being the largest zero of the $n$-th ultraspherical polynomial $P_n^{(λ)}$. For every fixed $λ>-1/2$, the limit of the ratio of our upper and lower bounds for $1-x_{nn}^2(λ)$ does not exceed $1.6$. This paper is a continuation of [1].
Our interest lies in describing the zero behaviour of Gauss hypergeometric polynomials $F(-n,b; c; z)$ where $b$ and $c$ are arbitrary parameters. In general, this problem has not been solved and even when $b$ and $c$ are both real, the only cases that have been fully analyzed impose additional restrictions on $b$ and $c$. We review recent results that have been proved for the zeros of several classes of hypergeometric polynomials $F(-n,b; c; z)$ where $b$ and $c$ are real. We show that the number of real zeros of $F(-n,b; c; z)$ for arbitrary real values of the parameters $b$ and $c$, as well…
We study a new family of "classical" orthogonal polynomials, here called big -1 Jacobi polynomials, which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with differential operators of Dunkl-type. These polynomials can be obtained from the big $q$-Jacobi polynomials in the limit $q \to -1$. An explicit expression of these polynomials in terms of Gauss' hypergeometric functions is found. The big -1 Jacobi polynomials are orthogonal on the union of two symmetric intervals of the real axis. We show that the big -1 Jacobi polynomials can be obtained from the Bannai-Ito poly…
We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of continuous q-ultraspherical or continuous q-Jacobi polynomials. As special cases, we obtain simple closed formulas for staircase-type partitions.
A correlation function of the classical orthogonal polynomials is defined and determined. The correlation function obeys a second order difference equation in two variables. The correlation function for the Gegenbauer, Chebyshev and Legendre polynomials can be written as a 4F3 hypergeometric function. For the Jacobi polynomials the result is an F2 Appell function. For the Generalized Laguerre polynomials the result is a confluent hypergeometric function and for the Hermite polynomials there rests only a single term.
We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $2\times 2$ introduced by C. Calderón et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel-Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of …
It turned out that the partial sums $g_n(z) = \sum_{k=0}^n \frac{(a_1)_k ... (a_p)_k}{(b_1)_k ... (b_q)_k} \frac{z^k}{k!}$, of the generalized hypergeometric series ${}_p F_q(a_1,...,a_p; b_1,...,b_q;z)$, with parameters $a_j,b_l\in\mathbb{C}\backslash\{ 0,-1,-2,... \}$, are Sobolev orthogonal polynomials. The corresponding monic polynomials $G_n(z)$ are polynomials of $R_I$ type, and therefore they are related to biorthogonal rational functions. Polynomials $g_n$ possess a differential equation (in $z$), and a recurrence relation (in $n$). We study integral representations for $g_n$, and some…
Generalized $f$-Eulerian polynomials: zeros and hypergeometric representations with applications
本文研究了广义$f$-Eulerian多项式，建立了其零点实数性的充分条件，揭示了与广义超几何函数的联系，并据此分析了$d$-Narayana多项式与Jacobi-Piñeiro多项式的性质及零点位置。
Using the technique of the elliptic Frobenius determinant, we construct new elliptic solutions of the $QD$-algorithm. These solutions can be interpreted as elliptic solutions of the discrete-time Toda chain as well. As a by-product, we obtain new explicit orthogonal and biorthogonal polynomials in terms of the elliptic hypergeometric function ${_3}E_2(z)$. Their recurrence coefficients are expressed in terms of the elliptic functions. In the degenerate case we obtain the Krall-Jacobi polynomials and their biorthogonal analogs.
Explicit expressions for the Hahn multiple polynomials of type I, in terms of Kampé de Fériet hypergeometric series, are given. Orthogonal and biorthogonal relations are proven. Then, part of the Askey scheme for multiple orthogonal polynomials type I is completed. In particular, explicit expressions in terms of generalized hypergeometric series and Kampé de Fériet hypergeometric series, are given for the multiple orthogonal polynomials of type I for the Jacobi-Piñeiro, Meixner I, Meixner II, Kravchuk, Laguerre I, Laguerre II and Charlier families.
The recently found hypergeometric multiple orthogonal polynomials on the step-line by Lima and Loureiro are shown to be random walk polynomials. It is proven that the corresponding Jacobi matrix and its transpose, which are nonnegative matrices and describe higher recurrence relations, can be normalized to two stochastic matrices, dual to each other. Using the Christoffel-Darboux formula on the step-line and the Poincaré theory for non-homogeneous recurrence relations it is proven that both stochastic matrices are related by transposition in the large $n$ limit. These random walks are beyond b…
Let $F_{BC}(λ,k;t)$ be the Heckman-Opdam hypergeometric function of type BC with multiplicities $k=(k_1,k_2,k_3)$ and weighted half sum $ρ(k)$ of positive roots. We prove that $F_{BC}(λ+ρ(k),k;t)$ converges for $k_1+k_2\to\infty$ and $k_1/k_2\to \infty$ to a function of type A for $t\in\b R^n$ and $λ\in\b C^n$. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitio…
We derive a novel integral representations of Jacobi polynomials in terms of the Gauss hypergeometric function. Such representation is then used to give the explicit integral representation for the Heat kernel on the quantized Riemann sphere.
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with multiplicity one. The spectral analysis gives rise to a generalized Fourier transform with an explicit hypergeometric function as a kernel. Using Jacobi polynomials the operator $T$ can also be realized as a five-diagonal operator, hence leading to orthogonality relations for $2\times 2$-matrix-valued polynomials. These matrix-valued polynomials can be considere…
Zeros of generalized hypergeometric polynomials via finite free convolution. Applications to multiple orthogonality
本文利用有限自由卷积表示法研究了广义超几何多项式零点的弱渐近行为，推导了极限分布的S变换，并分析了多重正交多项式零点的交错性、单调性与渐近性。
We study the $q$-hypergeometric difference operator $L$ on a particular Hilbert space. In this setting $L$ can be considered as an extension of the Jacobi operator for $q^{-1}$-Al-Salam--Chihara polynomials. Spectral analysis leads to unitarity and an explicit inverse of a $q$-analog of the Jacobi function transform. As a consequence a solution of the Al-Salam--Chihara indeterminate moment problem is obtained.
Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the norms of the polynomials in terms of the norm of the constant polynomial.
We consider the ratio of two Gauss hypergeometric functions with real parameters shifted by arbitrary integers. We find a formula for the jump of this ratio over the branch cut in terms of a real hypergeometric polynomial, the beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral representations for such ratio when the asymptotic behaviour at unity is mild and the denominator does not vanish. Multiplying the ratio by the aforementioned polynomial, we obtain a function that belongs to a generalized Nevanlinna class for arbitrary …
Integral and hypergeometric representations for multiple orthogonal polynomials
本文为任意权函数数量的经典多重正交多项式，建立了I型Hahn多项式的显式超几何表达式，并利用留数定理和Mellin变换推导了Laguerre、Jacobi-Piñeiro和Hahn族的围道积分表示，进而得到显式超几何表示。
The classical orthogonal polynomials are usually defined by the Rodrigues' formula. This paper refers to a fractional extension of the classical Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk and Hahn polynomials. By means of the Caputo operator of fractional calculus, C-Hermite, C-Laguerre, C-Legndre and the C-Jacobi functions are defined and their representation in terms of the hypergeometric functions are provided. Also, by means of the Gray and Zhang fractional difference oparator, fractional Charlier, Meixner, Krawtchouk and Hahn functions are defined and their representation in…
Even Hypergeometric Polynomials and Finite Free Commutators
本文研究了偶次超几何多项式与有限自由交换子的性质，并探讨了它们在自由概率中的应用。
A hypergeometric basis for the Alpert multiresolution analysis
本文构建了用于阿尔佩特多分辨率分析的分段超几何多项式正交基，通过超几何函数的傅里叶变换和矩阵方程的递推关系，推导出新的正交关系和恒等式。
Multiplicative and Additive Finite Free Convolutions for q-Polynomials
研究q-有限自由卷积及其与q-超几何多项式族的相互作用，引入q-加性有限自由卷积，并证明其与q-乘性卷积结合可保持实根性和交错性，同时发展出将q-超几何函数乘积恒等式转化为卷积恒等式的系统方法。
Hypergeometric Mixed-Type Multiple Orthogonal Polynomials
本文在正实轴上为任意秩一权重矩阵构造了类雅可比与类拉盖尔两类超几何混合型多重正交形式，显式给出其分量、递推与分解，并通过合流极限联系相关系统。
Compactly supported, orthogonal, continuous piecewise polynomial multiresolution analysis
本文提出基于超几何函数的正交多分辨率分析中的尺度函数显式表示，推导了其梅林变换和傅里叶变换的闭式表达，并引入新的多分辨率分析，其尺度函数系数为有理数。
New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed persymmetric Jacobi matrix. The orthogonality relation and an explicit expression in terms of hypergeometric functions are also given. Special cases and connections with the para-Krawtchouk polynomials and the dual-Hahn polynomials are also discussed.
In this paper we present an addition to Askey's scheme of q-hypergeometric orthogonal polynomials involving classes of q-special functions which do not consist of polynomials only. The special functions are q-analogues of the Jacobi and Bessel function, and are Askey-Wilson functions, big q-Jacobi functions and little q-Jacobi functions and the corrresponding q-Bessel functions. The generalised orthogonality relations and the second order q-difference equations for these families are given. Limit transitions between these families are discussed. The quantum group theoretic interpretations are …
Applications of multiple orthogonal polynomials with hypergeometric moment generating functions
本文研究了三类对应Jacobi、Laguerre、Bessel推广的、矩生成函数为变参数超几何级数的多重正交多项式，给出显式公式与II型零点渐近分布，还介绍了其在随机矩阵乘积奇异值研究和超几何级数有理线性无关性证明中的应用。
Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a parabolic element of the Lie algebra $\mathfrak{su}(1,1)$. The Clebsch-Gordan coefficients for the tensor product representation of a positive and a negative discrete series representation of $\mathfrak{su}(1,1)$ are determined for the parabolic bases. They turn out to be multiples of Jacobi functions. From the interpretation of Laguerre polynomials and fun…
Recurrence relations and applications for the Maclaurin coefficients of squared and cubic hypergeometric functions
本文研究了高斯超几何函数平方和立方的Maclaurin系数的递推关系，并应用于函数单调性分析和Clausen公式的证明。
A semi-infinite weighted Hankel matrix with entries defined in terms of basic hypergeometric series is explicitly diagonalized as an operator on $\ell^{2}(\mathbb{N}_{0})$. The approach uses the fact that the operator commutes with a diagonalizable Jacobi operator corresponding to Al-Salam-Chihara orthogonal polynomials. Yet another weighted Hankel matrix, which commutes with a Jacobi operator associated with the continuous $q$-Laguerre polynomials, is diagonalized. As an application, several new integral formulas for selected quantum orthogonal polynomials are deduced. In addition, an open re…
We establish a new connection between moments of $n \times n$ random matrices $X_n$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s \in \mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden et al. [F. D. Cunden, F. Mezzadri, N. J. S…
In 1995 Magnus posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [-1,1] of the form $$ (1-x)^α(1+x)^β|x_0 - x|^γ\times a jump at x_0, $$ with $α, β, γ>-1$ and $x_0 \in (-1,1)$. We show rigorously that Magnus' conjecture is correct even in a more general situation, when the weight above has an extra factor, which is analytic in a neighborhood of [-1,1] and positive on the interval. The proof is based on the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problem characteriz…
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