arXiv subjects

Manuel Mañas

Publications and source records attributed to Manuel Mañas.

At least 19 recordsLinked to original sources

arXiv2026

Positive Bidiagonal Factorizations for Banded Markov Processes

An ordered positive bidiagonal factorization (PBF) is used to develop a spectral and probabilistic theory for Markov transition matrices of arbitrary finite bandwidth. The factors determine two families of mixed-type multiple orthogonal polynomials, an entrywise positive $q\times p$ matrix of measures, and a sequence of elementary death-or-stay and birth-or-stay transitions. This yields Karlin-McGregor formulas for transition probabilities, Green kernels, resolvents, potentials, and first-passage transforms without reversibility or block symmetrizability. Rational stochastic PBFs are characterized by ordered finite-urn experiments. Cyclic reorderings give Darboux intertwinings, while a factor-resolved continued fraction gives the first-return law. Grouping states produces a finite-phase quasi-birth-and-death process and a matrix continued fraction for return time and phase. For bounded continuous-time generators, uniformization preserves the spectral data. For unbounded rates, a conservative generator whose shifted leading truncations all admit scalar PBFs must be tridiagonal; nevertheless, $Q=V(T-I)$ preserves arbitrary fixed bandwidth and separates the embedded chain from the holding rates. The theory is explicit for mixed Piñeiro and Jacobi-like systems. For Piñeiro, the exact PBF region and larger structural-band positivity regions are obtained. Jacobi-like beta-convolution weights admit Gamma-factor cancellations, complete cancellation recovering Piñeiro. For $q\in\{2,3,4\}$, the strict ordering conditions give the only open PBF region, with further lower-dimensional cancellation strata. Rational $(3,2)$ examples provide all factors and the resulting Markov models.

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arXiv2026

Bessel Multiple Orthogonal Polynomials of Mixed Type

This article constructs a Bessel family of mixed-type multiple orthogonal polynomials for a $q\times p$ matrix weight on the unit circle. Unlike a rank-one product weight, this matrix has generic rank $\min\{q,p\}$ outside a finite subset of the circle. Its reciprocal-Gamma moments recover the multiple Bessel system when $q=1$ and the Bessel-like system of Wolfs when $p=1$. The same matrix is obtained as a scaled Markov-Stieltjes limit of a rank-one Jacobi system, although the interval measures themselves have no finite limit. For balanced near-diagonal indices, explicit formulas are obtained for the mixed $A$ and $B$ polynomial vectors. Their orthogonality and weak normality are proved, and componentwise strong normality is characterized. The components have terminating generalized hypergeometric representations; the $B$ components also admit finite Kampé de Fériet representations and a matrix Rodrigues-type formula. In the one-row reduction, the bivariate representation becomes a generalized hypergeometric polynomial governed by a reflected type-II multiple Hahn polynomial. Finite Gamma-Pochhammer formulas give the near-diagonal and step-line recurrence coefficients. The corresponding banded recurrence matrix has a bidiagonal Christoffel factorization: the lower factors are evaluated from transformed polynomial vectors, while the upper factors are expressed through finite tau determinants. When $q=1$, every Christoffel step remains within the multiple Bessel family, and Gamma-Vandermonde determinants yield the complete factorization.

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arXiv2026

Hypergeometric Mixed-Type Multiple Orthogonal Polynomials

Two hypergeometric families of mixed-type multiple orthogonal forms are constructed for rank-one $q\times p$ matrices of weights, with arbitrary $p$ and $q$: Jacobi and Laguerre I systems. Both normalized mixed forms are obtained explicitly for admissible near-diagonal multi-indices. The power-vector components are terminating generalized hypergeometric polynomials, while the hypergeometric-vector components are finite sums of such polynomials. In the Jacobi case, these sums are expressed as finite combinations of terminating Kampé de Fériet polynomials evaluated at $(x,1)$. For the mixed beta--Euler Laguerre I system, a finite triangular system relates the residues at finite poles to the terms generated by the Euler operator. Gamma-quotient Mellin formulas, Meijer $G$-representations, and Rodrigues formulas are derived for the complete mixed forms. On the step-line, the two biorthogonal systems satisfy dual recurrences governed by matrices with $p$ subdiagonals and $q$ superdiagonals. All recurrence coefficients are given by finite Gamma--Pochhammer expressions. Under normality and nonvanishing-pivot assumptions, Christoffel transformations and Gauss--Borel factorization yield bidiagonal factorizations of the recurrence matrices. The lower factors have closed Pochhammer formulas, while the upper factors are expressed through finite Christoffel tau-determinants or, equivalently, cross-ratios of shifted moment minors. Both Christoffel chains close explicitly in the mixed Piñeiro specialization.

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arXiv2026

Positive Bidiagonal Factorizations: Combinatorial Structure, Spectral Theory, and Matrix Continued Fractions

Prescribed positive bidiagonal factorizations of semi-infinite banded matrices reveal spectral, approximation, and integrable structures for arbitrary lower and upper bandwidths. Every cyclic permutation of the factors admits a normalized lower--upper positive factorization. Under suitable degree conditions, factor transfers induce matrix Christoffel transformations, linking the cyclic Darboux orbit to mixed-type Favard theory. Christoffel words determine minimum-height positive refactorizations and the cyclic products that reach them. The associated matrices have two block displacements, reducing to two diagonals for coprime bandwidths. Their finite nonzero spectra lie on a star, and planar networks yield positive radial Stieltjes moment sequences. In the compact radial case star-supported representing measures are characterized by conditions at the origin and negative fractional moments. For coprime bandwidths, a radial moment deformation gives determinant solutions of a sparse Lax hierarchy and synchronizes the cyclic Toda flows. Retaining the prescribed factor order yields matrix continued fractions with explicit Padé-type contact, denominator factorizations, backward evaluation, and error bounds. For bounded nonnegative factors, the convergents are monotone and converge to the least nonnegative solution in the admissible domain. Under the Favard identification, they approximate the mixed-type Weyl matrix. Unbounded factorizations retain the formal approximation; analytic convergence requires a closed realization and stability hypotheses, with an additional identification for measure-defined Weyl matrices. Piñeiro and Jacobi-like systems give applications, including global positive coefficientwise integrable solutions under the stated positivity and AT hypotheses.

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arXiv2026

An Askey Scheme for Jacobi Systems of Mixed Type: Laguerre Limits of the First and Second Kinds and Hermite Limits

An Askey-type confluence scheme is developed for Jacobi multiple orthogonal systems of mixed type, with $q$ row weights and $p$ column weights. Rescaling the Jacobi variable near $0$ and near $1$ produces Laguerre systems of the first and second kinds. A Hermite system with one Gaussian--gamma row and $q-1$ exponential convolutions is obtained from Laguerre of the first kind, directly from the second kind, and directly from Jacobi. Explicit limiting weights and orthogonal forms are given, together with convergence of their mixed moments. For each fixed index, the recurrence coefficients and the entries of the bidiagonal factors also converge in the limit to Laguerre of the first kind, under nonvanishing conditions. The Laguerre family of the second kind is described by additive gamma convolutions. A closed product formula for its moment determinants holds for arbitrary positive real shape parameters and near-diagonal row multi-indices, yielding explicit normality conditions. Its forms admit contour and Rodrigues representations with terminating hypergeometric components: a multiple Kampé de Fériet series for $A$, and at most $q+1$ Srivastava--Daoust series for each $B$-component when $q\ge2$. For the Hermite family, normality holds at every Laguerre-admissible index under the stated separation conditions, and these indices form an infinite set. Its components have finite Hermite expansions and terminating Srivastava--Daoust representations. The three routes give the same limiting moments and normalized components, the Jacobi one being a single limit and the other two iterated. An appendix treats several derivative rows, where the connection determinant vanishes beyond a finite index range, with exact rank loss and diverging components. The one-row reductions recover the classical multiple second-kind Laguerre and Hermite systems.

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arXiv2026

Piñeiro mixed orthogonal polynomials and bidiagonal factorization

We study mixed multiple orthogonal polynomials associated with rank-one matrices of measures and their recurrence relations. On the step line we identify the corresponding banded recurrence matrix through the Gauss--Borel factorization of the moment matrix. Elementary Christoffel transformations lead to a factorization of this recurrence matrix into lower and upper bidiagonal factors. For the mixed Piñeiro system we derive contour-integral and generalized hypergeometric representations and formulas for the recurrence and factorization coefficients.

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arXiv2026

An Askey-Type Confluence Scheme for Hahn-Like Multiple Orthogonality

We embed the Jacobi-like and Laguerre-like systems for ordinary type-I/type-II multiple orthogonality introduced by Wolfs into a single Askey-type confluence scheme. Applying the Bernstein transform simultaneously to the $q$ Jacobi-like weights produces a positive finite-lattice Hahn-like ancestor, represented by multiple beta integrals, reducing to the classical Hahn weight for $q=1$, and converging back to the continuous system. Parameter and scaling limits then yield Kravchuk-like, two Meixner-like, two Charlier-like, Jacobi-like, two Laguerre-like, and Hermite-like families. We realize every arrow through explicit hypergeometric orthogonality data. For each near-diagonal multi-index $1\leq |\mathbf m|\leq N$, the Hahn-like type-II polynomial is a terminating ${}_{q+2}F_{q+1}(1)$ series satisfying an exact inverse Bernstein identity. We construct the normalized type-I form and, under explicit separation and nonvanishing assumptions, recover its individual polynomial components by finite sums of terminating hypergeometric functions and prove uniqueness. Finite-pole terms regroup into terminating Kampé de Fériet blocks. Their limits are obtained by coefficient extraction and reconstruction at infinity on the unreflected Kravchuk-like and Meixner-II-like branches, and by finite Lauricella--Horn sector sums on the reflected branches. This yields sectorwise component confluence along the Kravchuk--Charlier, Meixner--Charlier, Meixner--Laguerre, and Charlier--Hermite arrows. For $q=1$ all formulas reduce to the classical families; for $q>1$ the two Meixner-like systems, and likewise the two Charlier-like systems, are inequivalent under permutation of normalized rows.

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arXiv2026

Bidiagonal Factorization of Banded Recursion Matrices for Mixed-Type Multiple Orthogonal Polynomials

Given a banded matrix $\mathscr{T}_N$ with $p$ subdiagonals and $q$ superdiagonals arising from the Gauss--Borel factorization $\mathscr{M}_N = \mathscr{L}_N^{-1}\mathscr{U}_N^{-1}$ of a moment matrix, this paper constructs explicitly its bidiagonal factorization \[ \mathscr{T}_N = L_1 \cdots L_p\, U_q \cdots U_1. \] Bidiagonal factorizations of this type are central to the study of oscillatory banded matrices and to the spectral Favard theorem for multiple orthogonal polynomials The factorization is obtained via Christoffel transformations of the moment matrix. Provided that the perturbed moment matrices $\mathscr{M}_{N,(b,0)}$ and $\mathscr{M}_{N,(0,a)}$ admit a Gauss--Borel factorization, each bidiagonal factor is a quotient of the corresponding Gauss--Borel factors: \[ U_b = \mathscr{U}_{N,(b,0)}^{-1}\mathscr{U}_{N,(b-1,0)}, \qquad L_a = \mathscr{L}_{N,(0,a-1)}\mathscr{L}_{N,(0,a)}^{-1}. \] Explicit Christoffel-type formulas for the entries of the bidiagonal factors are then derived in terms of certain tau-determinants evaluated at the origin: \[ U_{b,n} = -\frac{τ^B_{b-1,n}\,τ^B_{b,n+1}} {τ^B_{b-1,n+1}\,τ^B_{b,n}}, \qquad L_{a,n+1} = -\frac{τ^A_{a-1,n+2}\,τ^A_{a,n}} {τ^A_{a-1,n+1}\,τ^A_{a,n+1}}. \] As an illustration, the theory is applied to the recurrence matrices of multiple Hahn orthogonal polynomials. For two weights the tetradiagonal case is handled via contiguous hypergeometric relations; for three weights, i.e. the pentadiagonal case, the direct hypergeometric representations are required. In both cases fully explicit bidiagonal factorizations are obtained.

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arXiv2026

Unbounded banded matrices, shifted positive bidiagonal factorizations, and mixed-type multiple orthogonality

This work extends Favard-type spectral representations for banded matrices $T$ beyond the bounded setting. It assumes that, for every $N\in\mathbb N_0$, there exists a shift $s_N\ge 0$ such that the shifted truncation $A_N:= T^{[N]}+s_N I_{N+1}$ admits a positive bidiagonal factorization (PBF). Allowing $s_N$ to depend on $N$ leads to a natural recentering step: the discrete Gauss-type quadrature measures associated with $A_N$ are translated by $x\mapsto x-s_N$, producing a uniformly bounded family of distribution functions. Combining moment stabilization for banded truncations with Helly-type compactness theorems yields a limiting matrix-valued measure, together with a Favard-type spectral representation and the corresponding mixed-type multiple biorthogonality relations. As a consequence, the classical Favard theorem for (possibly unbounded) Jacobi matrices is recovered as a special case. Indeed, for a tridiagonal $J$ with positive sub- and superdiagonals, each truncation $J^{[N]}$ admits a shift $s_N\ge 0$ such that $J^{[N]}+s_N I_{N+1}$ is oscillatory and therefore admits a PBF. The preceding construction then produces the usual spectral measure for $J$.

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arXiv2026

Spectral theory for Markov chains with transition matrix admitting a stochastic bidiagonal factorization

The recently established spectral Favard theorem for bounded banded matrices admitting a positive bidiagonal factorization is applied to a broader class of Markov chains with bounded banded transition matrices, extending beyond the classical birth-and-death setting, to those that allow a positive stochastic bidiagonal factorization. In the finite case, the Karlin-McGregor spectral representation is derived. The recurrence of the Markov chain is established, and explicit formulas for the stationary distributions are provided in terms of orthogonal polynomials. Analogous results are obtained for the countably infinite case. In this setting, the chain is not necessarily recurrent, and its behavior is characterized in terms of the associated spectral measure. Finally, ergodicity is examined through the presence of a mass at $1$ in the spectral measure, corresponding to the eigenvalue $1$ with both right and left eigenvectors.

math.PR↗

arXiv2025

Bidiagonal matrix factorisations related to multiple orthogonal polynomials

We provide necessary and sufficient conditions for the Hessenberg recurrence matrix associated with a system of multiple orthogonal polynomials to admit a factorisation as a product of bidiagonal matrices. Using the Gauss-Borel factorisation of the moment matrix, we show that the nontrivial entries of those bidiagonal matrices can be expressed in terms of coefficients of type I or type II multiple orthogonal polynomials on the step-line with respect to the original system and its Christoffel transformations. Using the connection of multiple orthogonal polynomials with branched continued fractions, we show that the nontrivial entries of the bidiagonal matrices in the factorisation of the Hessenberg recurrence matrix correspond to the coefficients of a branched continued fraction associated with the given system of multiple orthogonal polynomials. As a case study, we present an explicit bidiagonal factorisation for the Hessenberg recurrence matrices of the Jacobi-Piñeiro polynomials and, as a limiting case, the multiple Laguerre polynomials of first kind.

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arXiv2025

Bivariate multiple orthogonal polynomials of mixed type on the step-line

This article studies bivariate multiple orthogonal polynomials of the mixed type on the step-line. The analysis is based on the LU factorization of a moment matrix specifically adapted to this framework. The orthogonality and biorthogonality relations satisfied by these polynomials are identified, and their precise multi-degrees are determined. The corresponding recurrence relations and the growing band matrices that encode them are also derived. Christoffel-Darboux kernels and the associated Christoffel-Darboux-type formulas are obtained. An ABC-type theorem is established, relating the inverse of the truncated moment matrix to these kernels. As an illustration, the bivariate Jacobi-Piñeiro multiple orthogonal polynomials of mixed type on the triangle are computed by means of an LU factorization implemented in a dedicated Maple script.

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arXiv2025

Uvarov Perturbations for Multiple Orthogonal Polynomials of the Mixed Type on the Step-Line

Uvarov-type perturbations for mixed-type multiple orthogonal polynomials on the step line are investigated within a matrix-analytic framework. The transformations considered involve both rational and additive modifications of a rectangular matrix of measures, implemented through left and right multiplication by regular matrix polynomials together with the addition of finitely many discrete matrix masses. These operations induce structured finite-band modifications of the corresponding moment matrix and rational transformations of the associated Markov-Stieltjes matrix function. Explicit Uvarov-type connection formulas are obtained, relating the perturbed and unperturbed families of mixed-type multiple orthogonal polynomials, and determinant conditions ensuring the existence of the new orthogonality are established. The algebraic consequences of these transformations are analyzed, showing their effect on the structure of the block-banded recurrence operators and on the spectral data encoded in the Markov-Stieltjes matrix functions. As an illustrative example, nontrivial Uvarov-type perturbations of the Jacobi-Piñeiro multiple orthogonal polynomials are presented. The results unify Christoffel, Geronimus, and Uvarov transformations within a single matrix framework, providing a linear-algebraic perspective on rational and additive spectral transformations of structured moment matrices and their corresponding recurrence operators.

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arXiv2025

Matrix Bessel Biorthogonal Polynomials: A Riemann-Hilbert approach

We consider matrix orthogonal polynomials related to Bessel type matrices of weights that can be defined in terms of a given matrix Pearson equation. From a Riemann-Hilbert problem we derive first and second order differential relations for the matrix orthogonal polynomials and functions of second kind. It is shown that the corresponding matrix recurrence coefficients satisfy a non-Abelian extensions of a family of discrete Painlevé d-PIV equations. We present some nontrivial examples of matrix orthogonal polynomials of Bessel type.

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arXiv2024

General Geronimus Perturbations for Mixed Multiple Orthogonal Polynomials

General Geronimus transformations, defined by regular matrix polynomials that are neither required to be monic nor restricted by the rank of their leading coefficients, are applied through both right and left multiplication to a rectangular matrix of measures associated with mixed multiple orthogonal polynomials. These transformations produce Christoffel-type formulas that establish relationships between the perturbed and original polynomials. Moreover, it is proven that the existence of Geronimus-perturbed orthogonality is equivalent to the non-cancellation of certain $τ$-determinants. The effect of these transformations on the Markov-Stieltjes matrix functions is also determined. As a case study, we examine the Jacobi-Piñeiro orthogonal polynomials with three weights.

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arXiv2024

Mixed Multiple Orthogonal Laurent Polynomials on the Unit Circle

Mixed orthogonal Laurent polynomials on the unit circle of CMV type are constructed utilizing a matrix of moments and its Gauss--Borel factorization and employing a multiple extension of the CMV ordering. A systematic analysis of the associated multiple orthogonality and biorthogonality relations, and an examination of the degrees of the Laurent polynomials is given. Recurrence relations, expressed in terms of banded matrices, are found. These recurrence relations lay the groundwork for corresponding Christoffel-Darboux kernels and relations, as well as for elucidating the ABC theorem. The paper also develops the theory of diagonal Christoffel and Geronimus perturbations of the matrix of measures. Christoffel formulas are found for both perturbations.

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arXiv2024

Bidiagonal factorization of recurrence banded matrices in mixed multiple orthogonality

This paper demonstrates how to explicitly construct a bidiagonal factorization of the banded recurrence matrix that appears in mixed multiple orthogonality on the step-line in terms of the coeffcients of the mixed multiple orthogonal polynomials. The construction is based on the $LU$ factorization of the moment matrix and Christoffel transformations applied to the matrix of measures and the associated mixed multiple orthogonal polynomials.

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arXiv2024

Classical discrete multiple orthogonal polynomials: hypergeometric and integral representations

This work explores classical discrete multiple orthogonal polynomials, including Hahn, Meixner of the first and second kinds, Kravchuk, and Charlier polynomials, with an arbitrary number of weights. Explicit expressions for the recursion coefficients of Hahn multiple orthogonal polynomials are derived. By leveraging the multiple Askey scheme and the recently discovered explicit hypergeometric representation of type I multiple Hahn polynomials, corresponding explicit hypergeometric representations are provided for the type I polynomials and recursion coefficients of all the aforementioned descendants within the Askey scheme. Additionally, integral representations for these families within the Hahn class in the Askey scheme are presented. The multiple Askey scheme is further completed by providing the corresponding limits for the weights, polynomials, and recurrence coefficients.

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