# Roberto S. Costas-SantosUniversidad de Loyola Andalucia · Quantitative methods

Roberto S. Costas-Santos

PhD.

## About

50

Publications

2,851

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174

Citations

Introduction

Education

February 2003 - April 2007

September 1993 - August 1998

## Publications

Publications (50)

In this paper, we explore the symmetric nature of the terminating basic hypergeometric series representations of the Askey–Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. In particular we identify and classify the set of 4 and 7 equivalence classes of terminating balanced ϕ34...

We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were als...

In many cases one may encounter an integral which is of $q$-Mellin--Barnes type. These integrals are easily evaluated using theorems which have a long history dating back to Slater, Askey, Gasper, Rahman and others. We derive some interesting $q$-Mellin--Barnes integrals and using them we derive transformation and summation formulas for nonterminat...

In this contribution we consider sequences of monic polynomials orthogonal with respect to the Sobolev-type inner product f,g=⟨uM,fg⟩+λTjf(α)Tjg(α), where uM is the Meixner linear operator, λ∈R+, j∈N, α≤0, and T is the forward difference operator Δ or the backward difference operator ∇. Moreover, we derive an explicit representation for these polyn...

Using the direct relation between the Gegenbauer polynomials Cnλ(x) and the Ferrers function of the first kind Pνμ(x), we compute interrelations between certain Jacobi polynomials, Meixner polynomials, and Ferrers functions of the first and second kind. We then compute Rodrigues-type, standard integral orthogonality and Sobolev orthogonality relati...

In this paper, we explore the symmetric nature of the terminating basic hypergeometric series representations of the Askey--Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. In particular we identify and classify the set of 4 and 7 equivalence classes of terminating balanced ${...

Using the direct relation between the Gegenbauer polynomials and the Ferrers function of the first kind, we compute interrelations between certain Jacobi polynomials, Meixner polynomials, and the Ferrers function of the first kind. We then compute Rodrigues-type and orthogonality relations for Ferrers functions of the first and second kinds. In the...

We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were als...

The authors wish to make the following corrections to their paper [...]

In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product \[ \left\langle f,g\right\rangle _{\lambda,\mu}\!=\!\sum_{x=0}^Nf(x)g(x)\frac{\Gamma(N+1) p^x(1-p)^{N-x} }{\Gamma (N-x+1) \Gamma(x+1) }+\lambda\Delta^j f(0)\Delta^j g(0)+\mu\Delta^j f(N)\Delta^j g(N), \] where $0<p <1$, $\lambda,\mu...

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute...

In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey–Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy.

In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey-Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. From the terminating basic hypergeometric representations of these polynomials, and due to symmetry in their free pa...

We derive a generalization of the Rogers generating function for the continuous q-ultraspherical/Rogers polynomials whose coefficient is a ϕ12. From that expansion, we derive corresponding specialization and limit transition expansions for the continuous q-Hermite, continuous q-Legendre, Laguerre, and Chebyshev polynomials of the first kind. Using...

In this contribution, we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product ⟨f,g⟩S:=⟨u,fg⟩+N(Dqf)(α)(Dqg)(α),α∈R,N≥0, where u is a q-classical linear functional and Dq is the q-derivative operator. We obtain some algebraic properties of these polynomials such as an explicit representation, a five-term re...

In this contribution we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product \[ \langle f,g \rangle _{S}:= \langle {\bf u}, f g\rangle +N (\mathscr D_q f)(\alpha) (\mathscr D _{q}g)(\alpha),\qquad \alpha\in \mathbb R, \quad N\ge 0, \] where $\bf u$ is a $q$-classical linear functional and $\mathscr D _{q}$...

We consider spectra of $n$-by-$n$ irreducible tridiagonal matrices over a field and of their $n-1$-by-$n-1$ trailing principal submatrices. The real symmetric and complex Hermitian cases have been fully understood: it is necessary and sufficient that the necessarily real eigenvalues are distinct and those of the principal submatrix strictly interla...

We derive a generalized Rogers generating function and corresponding definite integral, for the continuous $q$-ultraspherical polynomials by applying its connection relation and utilizing orthogonality. Using a recent generalization of the Rogers generating function by Ismail & Simeonov expanded in terms of Askey-Wilson polynomials, we derive corre...

In this chapter, we study the orthogonality conditions satisfied by Al-Salam-Carlitz polynomials U(a)n (x; q) when the parameters a and q are not necessarily real nor "classical", i.e., the linear functional u with respect to such a polynomial sequence is quasi-definite and not positive definite. We establish orthogonality on a simple contour in th...

Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain q-analogues for some of their generating functions. Our q-generating functions are given in terms of the basic hypergeometric series 4 ϕ 5 , 5 ϕ 5 , 4 ϕ 3 , 3 ϕ 2 , 2 ϕ...

If T is a labelled tree, a matrix A is totally positive relative to T, principal submatrices of A associated with deletion of pendent vertices of T are P-matrices, and A has positive determinant, then the smallest absolute eigenvalue of A is positive with multiplicity 1 and its eigenvector is signed according to T. This conclusion has been incorrec...

En este estudio nos proponemos indagar si se pueden detectar evidencias para afir-mar que el género del individuo influye al obtener mejor resultado en una tarea matemática de orden mental. La revisión realizada acerca del tema muestra una mejor ejecución por parte de los hombres. Sin embargo, en los estudios también se reflejan las influencias am-...

In this contribution, we study the orthogonality conditions satisfied by Al-Salam-Carlitz polynomials $U^{(a)}_n(x;q)$ when the parameters $a$ and $q$ are not necessarily real nor `classical', i.e., the linear functional $\bf u$ with respect to such polynomial sequence is quasi-definite and not positive definite. We establish orthogonality on a sim...

We introduce the power collection method for easily deriving connection relations for certain hypergeometric orthogonal polynomials in the $(q-)$Askey scheme. We summarize the full-extent to which the power collection method may be used. As an example, we use the power collection method to derive connection and connection-type relations for Meixner...

We derive generalized generating functions for basic hypergeometric
orthogonal polynomials by applying connection relations with one free parameter
to them. In particular, we generalize generating functions for Askey-Wilson,
Rogers/continuous $q$-ultrapherical, little $q$-Laguerre/Wall, and $q$-Laguerre
polynomials. Depending on what type of orthog...

From the Rodrigues representation of polynomial eigenfunctions of a second order linear hypergeometric-type differential (difference or q-difference) operator, complementary polynomials for classical orthogonal polynomials are constructed using a straightforward method. Thus a generating function in a closed form is obtained. For the complementary...

In this paper we consider the polynomial sequence $(P_{n}^{\alpha,q}(x))$
that is orthogonal on $[-1,1]$ with respect to the weight function
$x^{2q+1}(1-x^{2})^{\alpha}(1-x), \alpha>-1, q\in \mathbb N$; we obtain the
coefficients of the tree-term recurrence relation (TTRR) by using a different
method from the one derived in \cite{kn:atia1}; we prov...

In this paper, we consider a natural extension of several results related to
Krall-type polynomials introducing a modification of a $q$-classical linear
functional via the addition of one or two mass points. The limit relations
between the $q$-Krall type modification of big $q$-Jacobi, little $q$-Jacobi,
big $q$-Laguerre, and other families of the...

q-polynomials can be defined for all the possible parameters, but their orthogonality properties are unknown for several configurations of the parameters. Indeed, orthogonality for the Askey–Wilson polynomials, pn(x;a,b,c,d;q), is known only when the product of any two parameters a,b,c,d is not a negative integer power of q. Also, the orthogonality...

06Bxx Lattices (See also 03G10)
39Axx Difference equations (For dynamical systems, see 37-XX)
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations)

We argue that one can factorize the difference equation of hypergeometric type on non-uniform lattices in the general case. It is shown that in the most cases of q-linear spectrum of the eigenvalues, this directly leads to the dynamical symmetry algebra suq(1, 1), whose generators are explicitly constructed in terms of the difference operators, obt...

We say that the polynomial sequence is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product where is a semiclassical linear functional, D is the differential, the difference or the q-difference operator, and λ is a positive constant.In this paper we get algebraic and differential/difference properties...

We discuss factorization of the hypergeometric-type difference equations on the uniform lattices and show how one can construct a dynamical algebra, which corresponds to each of these equations. Some examples are exhibited, in particular, we show that several models of discrete harmonic oscillators, previously considered in a number of publications...

In this paper we prove that for a general tree $T$, if $A$ is T-TP, all the submatrices of $A$ associated with the deletion of pendant vertices are $P$-matrices, and $\det A>0$, then the smallest eigenvalue has an eigenvector signed according to $T$. Comment: 6 pages, submitted to ELA

In this paper we study the orthogonality conditions satisfied by the
classical q-orthogonal polynomials that are located at the top of the q-Hahn
tableau (big q-jacobi polynomials (bqJ)) and the Nikiforov-Uvarov tableau
(Askey-Wilson polynomials (AW)) for almost any complex value of the parameters
and for all non-negative integers degrees. We state...

The q-classical orthogonal polynomials of the q-Hahn Tableau are characterized from their orthogonality condition and by a first and a second structure relation. Unfortunately, for the q-semiclassical orthogonal polynomials (a generalization of the classical ones) we find only in the literature the first structure relation. In this paper, a second...

We can write the polynomial solution of the second order linear differential equation of hypergeometric-type $$ \phi(x)y''+\psi(x)y'+\lambda y=0, $$ where $\phi$ and $\psi$ are polynomials, $\deg \phi\le 2$, $\deg \psi=1$ and $\lambda$ is a constant, among others, by using the Rodrigues operator $R_k(\phi,{\bf u})$ (see \cite{coma2}) where $\bf u$...

In this paper we find new integral representations for the {\it generalized Hermite linear functional} in the real line and the complex plane. As application, new integral representations for the Euler Gamma function are given.

It is well-known that the family of Hahn polynomials is orthogonal with respect to a certain weight function up to degree N. In this paper we prove, by using the three-term recurrence relation which this family satisfies, that the Hahn polynomials can be characterized by a Δ-Sobolev orthogonality for every n and present a factorization for Hahn pol...

In this paper we prove that for a general tree $T$, if $A$ is T-TP, all the
submatrices of $A$ associated with the deletion of pendant vertices are
$P$-matrices, and $\det A>0$, then the smallest eigenvalue has an eigenvector
signed according to $T$.

Often in mathematics it is useful to summarize a multivariate phenomenon with a single number and in fact, the determinant -- which is represented by det -- is one of the simplest cases. In fact, this number it is defined only for square matrices and a lot of its properties are very well-known. For instance, the determinant is a multiplicative func...

It is well known that the classical families of orthogonal polynomials are characterized as the polynomial eigenfunctions of a second order homogeneous linear differential/difference hypergeometric operator with polynomial coefficients.
In this paper we present a study of the classical orthogonal polynomials sequences, in short classical OPS, in a...

We study some q-analogs of Racah polynomials and some of their applications in the theory of representation of quantum algebras. Possible
implementations in quantum optics are discussed.

The question is raised whether the sum of the k×k principal minors of the titled matrix is a polynomial (in t) with positive coefficients, when A and B are positive definite. This would generalize a conjecture made by D. Bessis, P. Moussa, and M. Villani [J. Math. Phys. 16, 2318–2325 (1975; Zbl 0976.82501)], as stated by E. H. Lieb and R. Seiringer...

We study some q-analogues of the Racah polynomials and some of their
applications in the theory of representation of quantum algebras.

We study the factorization of the hypergeometric-type dierence equation of Nikiforov and Uvarov on nonuniform lattices. An explicit form of the raising and lowering operators is derived and some relevant examples are given. 1

La Teoría de funciones especiales y, más concretamente, la Teoría de polinomios ortogonales constituyen unas de las fuentes más apreciadas por la cantidad de aplicaciones tanto en la matemática y como en la física con la que éstas aparecen relacionadas. Entre ellas se encuentran la teoría de números, el análisis numérico, la teoría de operadores, l...

## Projects

Projects (5)

The goal of this project is to obtain algebraic and analytic properties for the sequentially ordered discrete Sobolev-type OP as well as for the balanced sequentially ordered discrete Sobolev-type OP.