A Jacobi type Christoffel-Darboux formula for multiple orthogonal polynomials of mixed type

An alternative expression for the Christoffel-Darboux formula for multiple orthogonal polynomials of mixed type is derived from the LU factorization of the moment matrix of a given measure and two sets of weights. We use the action of the generalized Jacobi matrix J, also responsible for the recurrence relations, on the linear forms and their duals to obtain the result

Lecture Notes in Complex Analysis

Apuntes de clase de análisis en una variable compleja para físicos. Se estudian funciones holomorfas, integrales complejas, series complejas, y la teoría de los residuos. Los elementos clave son las ecuaciones de Cauchy-Rieman, el teorema de Cauchy, la formula integral de Cauchy, y el teorema de los residuos. También, se incluye información histórica.Lecture notes on analysis in a complex variable for physicists. Holomorphic functions, complex integrals, complex series, and the theory of residues are studied. The key elements are the Cauchy-Rieman equations, Cauchy's theorem, Cauchy's integral formula, and the remainder theorem. Historical information is included as well.Depto. de Física TeóricaFac. de Ciencias FísicasFALSEunpu

Pearson equations for discrete orthogonal polynomials: III—Christoffel and Geronimus transformations

Contiguous hypergeometric relations for semiclassical discrete orthogonal polynomials are described as Christoffel and Geronimus transformations. Using the Christoffel–Geronimus– Uvarov formulas quasi-determinantal expressions for the shifted semiclassical discrete orthogonal polynomials are obtained

Crum transformation and Wronskian type solutions for supersymmetric KdV equation

©1997 Elsevier Science B.V. We should like to thank Allan Fordy and Kimio Ueno for making the reference [ 15] available to us.Darboux transformation is reconsidered for the supersymmetric KdV system. By iterating the Darboux transformation, a supersymmetric extension of the Crum transformation is obtained for the Manin-Radul SKdV equation, in doing so one gets Wronskian superdeterminant representations for the solutions. Particular examples provide us explicit supersymmetric extensions, super solitons, of the standard soliton of the KdV equation. The KdV soliton appears as the body of the super soliton.Depto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Vectorial Darboux transformations for the Kadomtsev-Petviashvili hierarchy

We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian type determinants. We also study the n-th Gel'fand-Dickey hierarchy introducing spectral operators and obtaining similar results. We reduce the above-mentioned results to the Kadomtsev-Petviashvili I and II real forms, obtaining corresponding vectorial Darboux transformations. In particular for the Kadomtsev-Petviashvili I hierarchy, we get the line soliton, the lump solution, and the Johnson-Thompson lump, and the corresponding determinant formulae for the nonlinear superposition of several of them. For Kadomtsev-Petviashvili II apart from the line solitons, we get singular rational solutions with its singularity set describing the motion of strings in the plane. We also consider the I and II real forms for the Gel'fand-Dickey hierarchies obtaining the vectorial Darboux transformation in both cases

The q-deformed mKP hierarchy with self-consistent sources, Wronskian solutions and solitons

Based on the eigenfunction symmetry constraint of the q-deformed modified KP hierarchy, a q-deformed mKP hierarchy with self-consistent sources (q-mKPHSCSs) is constructed. The q-mKPHSCSs contain two types of q-deformed mKP equation with self-consistent sources. By the combination of the dressing method and the method of variation of constants, a generalized dressing approach is proposed to solve the q-deformed KP hierarchy with self-consistent sources (q-KPHSCSs). Using the gauge transformation between the q-KPHSCSs and the q-mKPHSCSs, the q-deformed Wronskian solutions for the q-KPHSCSs and the q-mKPHSCSs are obtained. The one-soliton solutions for the q-deformed KP (mKP) equation with a source are given explicitly

Riemann–Hilbert problem and matrix biorthogonal polynomials

Recently the Riemann-Hilbert problem, with jumps supported on appropriate curves in the complex plane, has been presented for matrix biorthogonal polynomials, in particular non-Abelian Hermite matrix biorthogonal polynomials in the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coe cients first order matrix polynomials. We will explore this discussion, present some achievements and consider some new examples of weights for matrix biorthogonal polynomials.publishe

Christoffel transformation for a matrix of Bi-variate measures.

We consider the sequences of matrix bi-orthogonal polynomials with respect to the bilinear forms ((R) over cap) and ((L) over cap) (L) over cap = (TxT)integral P(z(1))L(z(1))d mu(z(1), z(2))Q(z(2)), where mu(z1, z2) is a matrix of bi-variate measures supported on T x T, with T the unit circle, L pxp[ z] is the set of matrix Laurent polynomials of size p x p and L(z) is a special polynomial in L pxp[ z]. A connection formula between the sequences of matrix Laurent bi-orthogonal polynomials with respect to ((R) over cap) and resp ((L) over cap) and the sequence of matrix Laurent bi-orthogonal polynomials with respect to d mu(z(1), z(2)) is given

Multivariate orthogonal Laurent polynomials and integrable systems

An ordering for Laurent polynomials in the algebraic torus (C*)(D), inspired by the Cantero-Moral-Velazquez approach to orthogonal Laurent polynomials in the unit circle, leads to the construction of a moment matrix for a given Borel measure in the unit torus T-D. The Gauss-Borel factorization of this moment matrix allows for the construction of multivariate biorthogonal Laurent polynomials in the unit torus, which can be expressed as last quasi-determinants of bordered truncations of the moment matrix. The associated second-kind functions are expressed in terms of the Fourier series of the given measure. Persymmetries and partial persymmetries of the moment matrix are studied and Cauchy integral representations of the second-kind functions are found, as well as Plemelj-type formulae. Spectral matrices give string equations for the moment matrix, which model the three-term relations as well as the Christoffel-Darboux formulae. Christoffel-type perturbations of the measure given by the multiplication by Laurent polynomials are studied. Sample matrices on poised sets of nodes, which belong to the algebraic hypersurface of the perturbing Laurent polynomial, are used to find a Christoffel formula that expresses the perturbed orthogonal Laurent polynomials in terms of a last quasi-determinant of a bordered sample matrix constructed in terms of the original orthogonal Laurent polynomials. Poised sets exist only for prepared Laurent polynomials, which are analyzed from the perspective of Newton polytopes and tropical geometry. Then, an algebraic geometrical characterization of prepared Laurent polynomial perturbation and poised sets is given; full-column-rankness of the corresponding multivariate Laurent-Vandermonde matrices and a product of different prime prepared Laurent polynomials leads to such sets. Some examples are constructed in terms of perturbations of the Lebesgue-Haar measure. Discrete and continuous deformations of the measure lead to a Toda-type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations and vertex operators are found. Varying size matrix nonlinear partial difference and differential equations of two-dimensional Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows are connected with a Gauss-Borel factorization of the Jacobi-type matrices and its quasi-determinants allow for expressions for the multivariate orthogonal polynomials in terms of shifted quasi-tau matrices, which generalize those that relate the Baker functions with ratios of Miwa shifted r-functions in the one-dimensional scenario. It is shown that the discrete and continuous flows are deeply connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomtsev-Petviashvili-type system