Toda chain,Stieltjes function,and orthogonal polynomials

We discuss relations between the theory of orthogonal polynomials, Hankel determinants, and the unrestricted one-dimensional Toda chain. In particular, we show that the equations of motion for the Toda chain are equivalent to a Riccati equation for the Stieltjes function. We consider some examples of the Stieltjes function with an explicit (hypergeometric and elliptic) time dependence in detail. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 81–108, April, 2007.     

Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation

Transformations of the measure of orthogonality for orthogonal polynomials, namely Freud transformations, are considered. Jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations. Dynamics of the coefficients are described by generalized Toda equations. The classical Toda lattice equations are the simplest special case of dynamics of the coefficients under the Freud transformation of the measure of orthogonality. Also dynamics of Hankel determinants, its minors and other notions corresponding to the orthogonal polynomials are studied.     

Elliptic solutions of the Toda chain and a generalization of the Stieltjes–Carlitz polynomials

We construct new elliptic solutions of the restricted Toda chain. These solutions give rise to a new explicit class of orthogonal polynomials, which can be considered as a generalization of the Stieltjes–Carlitz elliptic polynomials. Relations between characteristic (i.e., positive definite) functions, Toda chain, and orthogonal polynomials are developed in order to derive the main properties of these polynomials. Explicit expressions are found for the recurrence coefficients and the weight function for these polynomials. In the degenerate cases of the elliptic functions, the modified Meixner polynomials and the Krall–Laguerre polynomials appear.     

Free fermions, <Emphasis Type="Italic">W</Emphasis>-algebras,and isomonodromic deformations

We consider the theory of multicomponent free massless fermions in two dimensions and use it to construct representations of W-algebras at integer Virasoro central charges. We define the vertex operators in this theory in terms of solutions of the corresponding isomonodromy problem. We use this construction to obtain some new insights into tau functions of the multicomponent Toda-type hierarchies for the class of solutions given by the isomonodromy vertex operators and to obtain a useful representation for tau functions of isomonodromic deformations.     

A family of hyperbolic spin Calogero‐Moser systems and the spin Toda lattices

In this paper, we continue to develop a general scheme to study a broad class of integrable systems naturally associated with the coboundary dynamical Lie algebroids. In particular, we present a factorization method for solving the Hamiltonian flows. We also present two important classes of new examples, a family of hyperbolic spin Calogero‐Moser systems and the spin Toda lattices. To illustrate our factorization theory, we show how to solve these Hamiltonian systems explicitly. © 2004 Wiley Periodicals, Inc.     

Statistically q‐deformed and tau‐deformed systems

Two classes of statistically deformed systems are known in literature. They are, respectively, the q‐deformed systems (Lavagno and Narayana Swamy, Phys Rev E 2002, 65, 036101) and the κ‐deformed systems (Kaniadakis and Scarfone, Physica A 2002, 305, 69). In this article, a new class, i.e., the tau‐deformed systems, is introduced. For each of these systems, a consistent thermodynamics may be developed. A summary of the main similarities between the thermodynamic properties of q‐deformed and tau‐deformed systems is presented. The deformation outlined in this article is radically different from the nonextensive Tsallis statistics, where the structure of the entropy is rather arbitrary deformed via the logarithmic function. In contrast, the theory of tau‐deformed systems is developed on a purely physical basis. However, one finally shows that the tau‐systems may be described by using a new form of deformed logarithmic function. © 2009 Wiley Periodicals, Inc. Complexity, 2010     

Characterization of semi-classical orthogonal polynomials on nonuniform lattices

We state and prove characterization theorem for semi-classical orthogonal polynomials on nonuniform lattices (quadratic lattices of a discrete or q-discrete variable). This theorem proves the equivalence between the four characterization properties, namely, the Pearson type equation for the linear functional, the strictly quasi-orthogonality of the derivatives, the structure relation, and the Riccati equation for the formal Stieltjes function. We give the classification of the semi-classical linear functional of class one on nonuniform lattice. Using the definition and the properties of the associated orthogonal polynomials, we prove that semi-classical orthogonal polynomials satisfy the second-order divided difference equation on nonuniform lattices.     

Isomonodromic tau function on the space of admissible covers

The isomonodromic tau function of the Fuchsian differential equations associated to Frobenius structures on Hurwitz spaces can be viewed as a section of a line bundle on the space of admissible covers. We study the asymptotic behavior of the tau function near the boundary of this space and compute its divisor. This yields an explicit formula for the pullback of the Hodge class to the space of admissible covers in terms of the classes of compactification divisors.     

Orthogonal Polynomials on a Bi-lattice

We investigate generalizations of the Charlier and the Meixner polynomials on the lattice ? and on the shifted lattice ?+1???. We combine both lattices to obtain the bi-lattice ???(?+1???) and show that the orthogonal polynomials on this bi-lattice have recurrence coefficients that satisfy a nonlinear system of recurrence equations, which we can identify as a limiting case of an (asymmetric) discrete Painlevé equation.     

Hypergeometric Functions as Infinite-Soliton Tau Functions

It is known that resonant multisoliton solutions depend on higher times and a set of parameters (integrals of motion). We show that soliton tau functions of the Toda lattice (and of the multicomponent Toda lattice) are tau functions of a dual hierarchy, where the higher times and the parameters (integrals of motion) exchange roles. The multisoliton solutions turn out to be rational solutions of the dual hierarchy, and the infinite-soliton tau functions turn out to be hypergeometric-type tau functions of the dual hierarchy. The variables in the dual hierarchies exchange roles. Soliton momenta are related to the Frobenius coordinates of partitions in the decomposition of rational solutions with respect to Schur functions. As an example, we consider partition functions of matrix models: their perturbation series is, on one hand, a hypergeometric tau function and, on the other hand, can be interpreted as an infinite-soliton solution. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 222–250, February, 2006.     

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szeg? polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows — connected with Darboux transformations. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szeg? polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.     

Matrix Valued Orthogonal Polynomials Arising from the Complex Projective Space

The main purpose of this paper is to display new families of matrix valued orthogonal polynomials satisfying second-order differential equations, obtained from the representation theory of U(n). Given an arbitrary positive definite weight matrix W(t) one can consider the corresponding matrix valued orthogonal polynomials. These polynomials will be eigenfunctions of some symmetric second-order differential operator D only for very special choices of W(t). Starting from the theory of spherical functions associated to the pair (SU(n+1), U(n)) we obtain new families of such pairs {W,D}. These depend on enough integer parameters to obtain an immediate extension beyond these cases.     

Bi-orthogonal systems on the unit circle, regular semi-classical weights and integrable systems — II

We derive the Christoffel–Geronimus–Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities {z₁,…,z_M} the bi-orthogonal system is known to be monodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel–Geronimus–Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system — the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota–Miwa equations are derived for the τ-functions or equivalently Toeplitz determinants of the system.     

On the Tau Function for the Schlesinger Equation of Isomonodromic Deformations

The tau function for the Schlesinger equation of isomonodromic deformations is represented as the result of successively applied elementary gauge transformations; this, in particular, suggests a simple proof for the Miwa theorem about the tau function.     

Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

In this work we characterize a high-order Toda lattice in terms of a family of matrix polynomials orthogonal with respect to a complex matrix measure. In order to study the solution of this dynamical system, we give explicit expressions for the Weyl function, generalized Markov function, and we also obtain, under some conditions, a representation of the vector of linear functionals associated with this system. We show that the orthogonality is embedded in these structure and governs the high-order Toda lattice. We also present a Lax-type theorem for the point spectrum of the Jacobi operator associated with a Toda-type lattice.     

The operational matrix of fractional integration for shifted Chebyshev polynomials

A new shifted Chebyshev operational matrix (SCOM) of fractional integration of arbitrary order is introduced and applied together with spectral tau method for solving linear fractional differential equations (FDEs). The fractional integration is described in the Riemann–Liouville sense. The numerical approach is based on the shifted Chebyshev tau method. The main characteristic behind the approach using this technique is that only a small number of shifted Chebyshev polynomials is needed to obtain a satisfactory result. Illustrative examples reveal that the present method is very effective and convenient for linear multi-term FDEs.     

Numerical integration over a semi-infinite interval,using the lognormal distribution

Summary A method is described for numerical integration over a semiinfinite interval using a Gaussian formula, with the corresponding set of orthogonal polynomials constructed from a lognormal weight function.The lognormal weight function and hence the coefficients of the polynomials are functions of two arbitrary parameters; the mean and the logarithmic variance. The method is found to be of particular use for integration of bell-shaped or sharply spiked functions. Rapidly convergent results can be obtained in these cases, since the lognormal distribution can be used to provide a good approximation to the actual function to be integrated, by suitable choice of the two arbitrary parameters. Two examples are given for integrals with known solutions.     

A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients

This paper presents a shifted fractional‐order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time‐fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.     

Characterizations of classical orthogonal polynomials on quadratic lattices

This paper is devoted to characterizations of classical orthogonal polynomials on quadratic lattices by using a matrix approach. In this form we recover the Hahn, Geronimus, Tricomi and Bochner type characterizations of classical orthogonal polynomials on quadratic lattices. Moreover a new matrix characterization of classical ortho-gonal polynomials in quadratic lattices is presented. From the Bochner type characterization we derive the three-term recurrence relation coefficients for these polynomials.     

Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights

The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject.In the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane. The strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a Riemann-Hilbert analysis.In the third part we use Strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves. As a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of Jordan arcs forming an arbitrary forest of trivalent loop-free trees.