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Roberto S. Costas-Santos

Publications and source records attributed to Roberto S. Costas-Santos.

At least 19 recordsLinked to original sources
arXiv2025

Terminating representations, transformations and summations for the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials

In this article, we exhaustively explore the terminating basic hypergeometric representations and transformations of the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials. These subfamilies are obtained by repeatedly setting one of the free parameters (not $q$) equal to zero until no parameters are left. These subfamilies (and their $q^{-1}$ counterparts) are the continuous dual $q$-Hahn, Al-Salam--Chihara, continuous big $q$-Hermite, and the continuous $q$-Hermite polynomials. From the terminating basic hypergeometric representations of these polynomials, and due to symmetry in their free parameters, we are able to exhaustively explore the terminating basic hypergeometric transformation formulas which these polynomials satisfy. We then study the terminating transformation structure which are implied by the terminating representations of these polynomials. We conclude by describing the symmetry group structure of the $q$-Askey scheme.

math.CA↗
arXiv2025

The $q$ and $q^{-1}$-symmetric orthogonal polynomials in the $q$-Askey scheme, their dual polynomials and functions, orthogonality, generating functions and relations and nonterminating $q$-Chaundy double product representations

We derive double-product representations of nonterminating basic hypergeometric series using diagonalization, a method introduced by Theo William Chaundy in 1943. We refer to this result as the $q$-Chaundy theorem and several limiting $q\to 1^{-}$ cases are considered. Using the $q$-Chaundy theorem, we explore properties of the symmetric and $q^{-1}$-symmetric basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. These are the continuous dual $q$ and $q^{-1}$-Hahn polynomials, the $q$ and $q^{-1}$-Al-Salam--Chihara polynomials, the continuous big $q$ and $q^{-1}$-Hermite polynomials and the continuous $q$ and $q^{-1}$-Hermite polynomials. For instance, we show how many known (and unknown) generating functions can be easily derived for these polynomials. We also explore other methods to find generating functions for these polynomials. By applying the $q$-Chaundy theorem to the Ismail--Masson $q$-exponential generating function for continuous $q$ and $q^{-1}$-Hermite polynomials, we are able to derive alternative expansions of these generating functions, and from these, new terminating basic hypergeometric representations for the continuous $q$ and $q^{-1}$-Hermite polynomials. New quadratic transformations for the terminating basic hypergeometric series involved connect these representations. For the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials and as well their dual polynomials, which include the big and little $q$-Jacobi polynomials and the $q^{-1}$-Bessel polynomials, we discuss, and show how to exploit special orthogonality relations (integral and infinite series), connection formulas, and duality relations for these infinite families to derive new generating relations, and as well summation and integration formulas.

math.CA↗
arXiv2025

The Laguerre constellation of classical orthogonal Polynomials

A linear functional $\bf u$ is classical if there exist polynomials, $ϕ$ and $ψ$, with $°ϕ\le 2$, $°ψ=1$, such that ${\mathscr D}\left(ϕ(x) {\bf u}\right)=ψ(x){\bf u}$, where ${\mathscr D}$ is a certain differential, or difference, operator. The polynomials orthogonal with respect to the linear functional ${\bf u}$ are called {\sf classical orthogonal polynomials}. In the theory of orthogonal polynomials, a correct characterization of the classical families is of great interest. In this work, on the one hand, we present the Laguerre constellation, which is formed by all the classical families for which $°ϕ=1$, obtaining for all of them new algebraic identities such as structure formulas, orthogonality properties as well as new Rodrigues formulas; on the other hand, we present a theorem that characterizes the classical families belonging to the Laguerre constellation.

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arXiv2024

Investigating Multidimensional Degenerate Hybrid Special Polynomials and Their Connection to Appell Sequences: Properties and Applications

This paper explores the operational principles and monomiality principle that significantly shape the development of various special polynomial families. We argue that applying the monomiality principle yields novel results while remaining consistent with established findings. The primary focus of this study is the introduction of degenerate multidimensional Hermite-based Appell polynomials (DMHAP), denoted as ${}_{\mathbb{H}}\mathbb{A}^{[r]}_n(l_1, l_2, l_3, \dots, l_r; \vartheta)$. These DMHAP form essential families of orthogonal polynomials, demonstrating strong connections with classical Hermite and Appell polynomials. Additionally, we derive symmetric identities and examine the fundamental properties of these polynomials. Finally, we establish an operational framework to investigate and develop these polynomials further.

math.GM↗
arXiv2024

Asymptotics, orthogonality relations and duality for the $q$ and $q^{-1}$-symmetric polynomials in the $q$-Askey scheme

In this survey we summarize the current state of known orthogonality relations for the $q$ and $q^{-1}$-symmetric and dual subfamilies of the Askey--Wilson polynomials in the $q$-Askey scheme. These polynomials are the continuous dual $q$ and $q^{-1}$-Hahn polynomials, the $q$ and $q^{-1}$-Al-Salam--Chihara polynomials, the continuous big $q$ and $q^{-1}$-Hermite polynomials and the continuous $q$ and $q^{-1}$-Hermite polynomials and their dual counterparts which are connected with the big $q$-Jacobi polynomials, the little $q$-Jacobi polynomials and the $q$ and $q^{-1}$-Bessel polynomials. The $q^{-1}$-symmetric polynomials in the $q$-Askey scheme satisfy an indeterminate moment problem, satisfying an infinite number of orthogonality relations for these polynomials. Among the infinite number of orthogonality relations for the $q^{-1}$-symmetric families, we attempt to summarize those currently known. These fall into several classes, including continuous orthogonality relations and infinite discrete (including bilateral) orthogonality relations. Using symmetric limits, we derive a new infinite discrete orthogonality relation for the continuous big $q^{-1}$-Hermite polynomials. Using duality relations, we explore orthogonality relations for and from the dual families associated with the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials. In order to give a complete description of the convergence properties for these polynomials, we provide the large degree asymptotics using the Darboux method for these polynomials. In order to apply the Darboux method, we derive a generating function with two free parameters for the $q^{-1}$-Al-Salam--Chihara polynomials which has natural limits to the lower $q^{-1}$-symmetric families.

math.CA↗
arXiv2024

On Polar Jacobi Polynomials

In the present work, we investigate certain algebraic and differential properties of the orthogonal polynomials with respect to a discrete-continuous Sobolev-type inner product defined in terms of the Jacobi measure.

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arXiv2023

Sobolev orthogonal polynomials: Connection formulae

This contribution aims to obtain several connection formulae for the polynomial sequence, which is orthogonal with respect to the discrete Sobolev inner product \[ \langle f, g\rangle_n=\langle {\bf u}, fg\rangle+ \sum_{j=1}^M μ_{j} f^{(ν_j)}(c_j) g^{(ν_j)}(c_j), \] where ${\bf u}$ is a classical linear functional, $c_j\in \mathbb R$, $ν_j\in \mathbb N_0$, $j=1, 2,...., M$. The Laguerre case will be considered.

math.CA↗
arXiv2023

Orthogonality of the big $-1$ Jacobi polynomials for non-standard parameters

The big $-1$ Jacobi polynomials $(Q_n^{(0)}(x;α,β,c))_n$ have been classically defined for $α,β\in(-1,\infty)$, $c\in(-1,1)$. We extend this family so that wider sets of parameters are allowed, i.e., they are non-standard. Assuming initial conditions $Q^{(0)}_0(x)=1$, $Q^{(0)}_{-1}(x)=0$, we consider the big $-1$ Jacobi polynomials as monic orthogonal polynomials which therefore satisfy the following three-term recurrence relation \[ xQ^{(0)}_n(x)=Q^{(0)}_{n+1}(x)+b_{n} Q^{(0)}_n(x)+ u_{n} Q^{(0)}_{n-1}(x), \quad n=0, 1, 2,\ldots. \] For standard parameters, the coefficients $u_n>0$ for all $n$. We discuss the situation where Favard's theorem cannot be directly applied for some positive integer $n$ such that $u_n=0$. We express the big $-1$ Jacobi polynomials for non-standard parameters as a product of two polynomials. Using this factorization, we obtain a bilinear form with respect to which these polynomials are orthogonal.

math.CA↗
arXiv2023

Multi-integral representations for Jacobi functions of the first and second kind

One may consider the generalization of Jacobi polynomials and the Jacobi function of the second kind to a general function where the index is allowed to be a complex number instead of a non-negative integer. These functions are referred to as Jacobi functions. In a similar fashion as associated Legendre functions, these break into two categories, functions which are analytically continued from the real line segment $(-1,1)$ and those continued from the real ray $(1, \infty)$. Using properties of Gauss hypergeometric functions, we derive multi-derivative and multi-integral representations for the Jacobi functions of the first and second kind.

math.CA↗
arXiv2023

Double summation addition theorems for Jacobi functions of the first and second kind

In this paper we review and derive hyperbolic and trigonometric double summation addition theorems for Jacobi functions of the first and second kind. In connection with these addition theorems, we perform a full analysis of the relation between symmetric, antisymmetric and odd-half-integer parameter values for the Jacobi functions with certain Gauss hypergeometric functions which satisfy a quadratic transformation, including associated Legendre, Gegenbauer and Ferrers functions of the first and second kind. We also introduce Olver normalizations of the Jacobi functions which are particularly useful in the derivation of expansion formulas when the parameters are integers. We introduce an application of the addition theorems for the Jacobi functions of the second kind to separated eigenfunction expansions of a fundamental solution of the Laplace-Beltrami operator on the compact and noncompact rank one symmetric spaces.

math.CA↗
arXiv2023

Special values for continuous $q$-Jacobi polynomials

We study special values for the continuous $q$-Jacobi polynomials and present applications of these special values which arise from bilinear generating functions, and in particular the Poisson kernel for these polynomials.

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arXiv2022

Utility of integral representations for basic hypergeometric functions and orthogonal polynomials

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 also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integrals connected to nonterminating basic hypergeometric series and the Askey--Wilson polynomials. We express a certain integral of a ratio of infinite $q$-shifted factorials as a symmetric sum of two basic hypergeometric series with argument $q$. The result is then expressed as a $q$-integral. Examples of integral representations applied to the derivation of generating functions for Askey--Wilson are given and as well the computation of a missing generating function for the continuous dual $q$-Hahn polynomials.

math.CA↗
arXiv2022

Nonterminating transformations and summations associated with some q-Mellin--Barnes integrals

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 nonterminating basic hypergeometric functions. The cases which we treat include ratios of theta functions, the Askey--Wilson moments, nonterminating well-poised ${}_3ϕ_2$, nonterminating very-well-poised ${}_5W_4$, ${}_8W_7$, products of two nonterminating ${}_2ϕ_1$'s, square of a nonterminating well-poised ${}_2ϕ_1$, and nonterminating ${}_{12}W_{11}$ and ${}_{10}W_9$.

math.CA↗
arXiv2022

Symmetry of terminating series representations of the Askey-Wilson polynomials

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 ${}_4ϕ_3$ and terminating very-well poised ${}_8W_7$ basic hypergeometric series which are connected with the Askey--Wilson polynomials. We study the inversion properties of these equivalence classes and also identify the connection of both sets of equivalence classes with the symmetric group $S_6$, the symmetry group of the terminating balanced ${}_4ϕ_3$. We then use terminating balanced ${}_4ϕ_3$ and terminating very-well poised ${}_8W_7$ transformations to give a broader interpretation of Watson's $q$-analog of Whipple's theorem and its converse. We give a broad description of the symmetry structure of the terminating basic hypergeometric series representations of the Askey--Wilson polynomials.

math.CA↗
arXiv2021

On the relation between Gegenbauer polynomials and the Ferrers function of the first kind

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 remainder of the paper using the relation between Gegenbauer polynomials and the Ferrers function of the first kind we derive connection and linearization relations, some definite integral and series expansions, some asymptotic expansions of Mehler-Heine type, Christoffel-Darboux summation formulas, and infinite series closure relations (Dirac delta distribution).

math.CA↗
arXiv2020

On difference equations of Kravchuk-Sobolev type polynomials of higher order

In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product \[ \left\langle f,g\right\rangle _{λ,μ}\!=\!\sum_{x=0}^Nf(x)g(x)\frac{Γ(N+1) p^x(1-p)^{N-x} }{Γ(N-x+1) Γ(x+1) }+λΔ^j f(0)Δ^j g(0)+μΔ^j f(N)Δ^j g(N), \] where $0<p <1$, $λ,μ\in \mathbb R_{+}$, $n\leq N\in \mathbb Z_{+}$, $j\in \mathbb Z_{+}$ and $Δ$ denotes the forward difference operators. We derive an explicit representation for these polynomials. In addition, the ladder operators associated with these polynomials are obtained. As a consequence, the linear difference equations of second order are also given.

math.CA↗
arXiv2020

Terminating basic hypergeometric representations and transformations for the Askey-Wilson polynomials

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 parameters, we are able to exhaustively explore the terminating basic hypergeometric transformation formulae which these polynomials satisfy.

math.CA↗