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Luis Verde-Star

Publications and source records attributed to Luis Verde-Star.

10 recordsLinked to original sources
arXiv2026

Classification of the hypergeometric orthogonal polynomials via their recurrence coefficients

We present a new classification of the class of all the hypergeometric orthogonal polynomial sequences. The classification uses properties of the coefficients $\alpha_n$ and $\beta_n$ in the three-term recurrence relation satisfied by the orthogonal polynomial sequences. Such coefficients are rational functions of $n$ and are determined by a set of six parameters. The partial fraction decomposition of $\alpha_n$ is a linear combination of five linearly independent functions of $n$ with coefficients $d_j$, for $0 \le j \le 4$, that are polynomials in our six parameters. For each value of a parameter $r$, associated with the eigenvalues, we classify the $\alpha_n$ according to the set of coefficients $d_j$ that are nonzero. There are certain particular values of $r$ that must be considered separately. Our classification yields a collection of 53 disjoint classes and it is different from the Askey scheme in several aspects. It is similar to the classifications proposed recently by Koornwinder.

math.CA↗
arXiv2024

$q$-Hypergeometric Orthogonal Polynomials with $q=-1$

We obtain some properties of a class $\mathcal{A}$ of $q$-hypergeometric orthogonal polynomials with $q=-1$, described by a uniform parametrization of the recurrence coefficients. We construct a class $\mathcal{C}$ of complementary $-1$ polynomials by means of the Darboux transformation with a shift. We show that our classes contain the Bannai-Ito polynomials and their complementary polynomials and other known $-1$ polynomials. We introduce some new examples of $-1$ polynomials and also obtain matrix realizations of the Bannai-Ito algebra.

math.CA↗
arXiv2024

A Linear Algebra approach to monomiality and operational methods

We use linear algebraic methods to obtain general results about linear operators on a space of polynomials that we apply to the operators associated with a polynomial sequence by the monomiality property. We show that all such operators are differential operators with polynomial coefficients of finite of infinite order. We consider the monomiality operators associated with several classes of polynomial sequences, such as Appell and Sheffer, and also orthogonal polynomial sequences that include the Meixner, Krawtchouk, Laguerre, Meixner-Pollaczek, and Hermite families.

math.CA↗
arXiv2023

Linearization and connection coefficients of polynomial sequences: A matrix approach

For a sequence of polynomials $\{p_k(t)\}$ in one real or complex variable, where $p_k$ has degree $k$, for $k\ge 0$, we find explicit expressions and recurrence relations for infinite matrices whose entries are the coefficients $d(n,m,k)$, called linearization coefficients, that satisfy $$ p_n(t) p_m(t)=\sum_{k=0}^{n+m} d(n,m,k) p_k(t).$$ For any pair of polynomial sequences $\{u_k(t)\}$ and $\{p_k(t)\}$ we find infinite matrices whose entries are the coefficients $e(n,m,k)$ that satisfy $$p_n(t) p_m(t)=\sum_{k=0}^{n+m} e(n,m,k) u_k(t).$$ Such results are obtained using a matrix approach. We also obtain recurrence relations for the linearization coefficients, apply the general results to general orthogonal polynomial sequences and to particular families of orthogonal polynomials such as the Chebyshev, Hermite, and Charlier families.

math.RA↗
arXiv2022

Discrete orthogonality of the polynomial sequences in the $q$-Askey scheme

We obtain weight functions associated with $q$-linear and $q$-quadratic lattices that yield discrete orthogonality with respect to a quasi-definite moment functional for the Askey-Wilson polynomials and all the polynomial sequences in the q-Askey scheme, with the exception of the continuous $q$-Hermite polynomials.

math.CA↗
arXiv2021

A unified construction of all the hypergeometric and basic hypergeometric families of orthogonal polynomial sequences

We construct a set $H$ of orthogonal polynomial sequences that contains all the families in the Askey scheme and the $q$-Askey scheme. The polynomial sequences in $H$ are solutions of a generalized first-order difference equation which is determined by three linearly recurrent sequences of numbers. Two of these sequences are solutions of a difference equation of order three and the other sequence satisfies a related difference equation of order five. We obtain explicit expressions for the coefficients of the orthogonal polynomials and for the generalized moments with respect to a basis of Newton type of the space of polynomials. We also obtain explicit formulas for the coefficients of the three-term recurrence relation satisfied by the polynomial sequences in $H$. The set $H$ contains all the 15 families in the Askey scheme and all the 29 families in the $q$-Askey scheme. Each of these families is obtained by direct substitution of appropriate values for the parameters in our general formulas. The only cases that require some limits are the Hermite and continuous $q$-Hermite polynomials. We present the values of the parameters for some of the families.

math.CA↗
arXiv2016

A class of q-orthogonal polynomial sequences that extends the q-Askey scheme

We obtain new explicit formulas for the recurrence coefficients of the q-orthogonal polynomial sequences in a class that extends the q-Askey scheme. Our formulas express the recurrence coefficients in terms of four parameters that determine the zeroes and poles. By direct substitution of particular values for the four parameters we obtain all the polynomial sequences in the q-Askey scheme, including the Askey-Wilson and the Racah polynomials. We also obtain some families of sequences that may be new.

math.CA↗
arXiv2010

An algorithm for the Cartan-Dieudonné theorem on generalized scalar product spaces

We present an algorithmic proof of the Cartan-Dieudonné theorem on generalized real scalar product spaces with arbitrary signature. We use Clifford algebras to compute the factorization of a given orthogonal transformation as a product of reflections with respect to hyperplanes. The relationship with the Cartan-Dieudonné-Scherk theorem is also discussed in relation to the minimum number of reflections required to decompose a given orthogonal transformation.

math.RA↗