arXiv subjects
Publications and source records attributed to R. S. Costas-Santos.
In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given.
In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product \[ \left\langle f,g\right\rangle= \langle {\bf u}^{\tt M},fg\rangle+λ\mathscr T^j f (α)\mathscr T^{j}g(α), \] where ${\bf u}^{\tt M}$ is the Meixner linear operator, $λ\in\mathbb{R}_{+}$, $j\in\mathbb{N}$, $α\leq 0$, and $\mathscr T$ is the forward difference operator $Δ$, or the backward difference operator $\nabla$. We derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equation of second order is also given. In addition, for these polynomials we derive a $(2j+3)$-term recurrence relation. Finally, we find the Mehler-Heine type formula for the $α\le 0$ case.
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 interlace. So this case is very restrictive. By contrast, for a general field, the requirements on the two spectra are much less restrictive. In particular, in the real or complex case, the $n$-by-$n$ characteristic polynomial is arbitrary (so that the algebraic multiplicities may be anything in place of all 1's in the classical cases) and that of the principal submatrix is the complement of a lower dimensional algebraic set (and so relatively free). Explicit conditions are given.
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 simple contour in the complex plane which depends on the parameters. In all cases we show that the orthogonality conditions characterize the Al-Salam-Carlitz polynomials $U_n^{(a)}(x;q)$ of degree $n$ up to a constant factor. We also obtain a generalization of the unique generating function for these polynomials.
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 $T$.
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 $q$-Hahn tableau are established.
We study the factorization of the hypergeometric-type difference 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.
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, can be treated in a unified form.
We say that the polynomial sequence $(Q^{(λ)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ _S=<{\bf u} ,{p\, r}> +λ<{\bf u}, {{\mathscr D}p \,{\mathscr D}r}>, $$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ 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 for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $\bf u$. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator ${\mathscr D}$ considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time.
We argue that one can factorize the difference equation of hypergeometric type on the nonuniform lattices in 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 $su_q(1,1)$, whose generators are explicitly constructed in terms of the difference operators, obtained in the process of factorization. Thus all models with the $q$-linear spectrum (some of them, but not all, previously considered in a number of publications) can be treated in a unified form.
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 function, i.e. det(AB)=detA detB, but it is not, in general, an additive function. Another interesting function in the matrix analysis is the characteristic polynomial -- in fact, given a matrix A, this function is defined by $p_A(t)=det(tI-A)$ where I is the identity matrix -- which elements are, up a sign, the elementary symmetric functions associated to the eigenvalues of the matrix A. In the present paper new expressions related with the determinant of sum of matrices and the elementary symmetric functions are given. Moreover, the connection with the Mobius function and the partial ordered sets (poset) is presented. Finally, a problem related with the determinant of sum of matrices is solved.
It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a more general context by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified representation of them. Furthermore, some well known results related to the Rodrigues operator are deduced. A more general characterization Theorem that the one given in [1] and [2] for the q-polynomials of the q-Askey and Hahn Tableaux, respectively, is established. Finally, the families of Askey-Wilson polynomials, q-Racah polynomials, Al-Salam & Carlitz I and II, and q-Meixner are considered. [1] R. Alvarez-Nodarse. On characterization of classical polynomials. J. Comput. Appl. Math., 196:320{337, 2006. [2] M. Alfaro and R. Alvarez-Nodarse. A characterization of the classical orthogonal discrete and q-polynomials. J. Comput. Appl. Math., 2006. In press.
It is well known that the family of Hahn polynomials $\{h_n^{α,β}(x;N)\}_{n\ge 0}$ is orthogonal with respect to a certain weight function up to $N$. In this paper we present a factorization for Hahn polynomials for a degree higher than $N$ and we prove that these polynomials can be characterized by a $Δ$-Sobolev orthogonality. We also present an analogous result for dual-Hahn, Krawtchouk, and Racah polynomials and give the limit relations between them for all $n\in \XX N_0$. Furthermore, in order to get this results for the Krawtchouk polynomials we will get a more general property of orthogonality for Meixner polynomials.
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 structure relation is deduced. In particular, by means of a general finite-type relation between a q-semiclassical polynomial sequence and the sequence of its q-differences such a structure relation is obtained.
We can write the polynomial solution of the second order linear differential equation of hypergeometric-type $$ ϕ(x)y''+ψ(x)y'+λy=0, $$ where $ϕ$ and $ψ$ are polynomials, $°ϕ\le 2$, $°ψ=1$ and $λ$ is a constant, among others, by using the Rodrigues operator $R_k(ϕ,{\bf u})$ (see \cite{coma2}) where $\bf u$ is certain linear operator which satisfies the distributional equation \begin{equation} \label{1} \frac{d}{dx}[ϕ{\bf u}]=ψ{\bf u}, \end{equation} as $$ P_n(x)=B_n R_n(ϕ,{\bf u})[1], \qquad B_n\ne 0,\quad n=0, 1, 2, ... $$ Taking this into account we construct the complementary polynomials. Among the key results is a generating functional function in closed form leading to derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and Rodrigues formulas.
We study some q-analogues of the Racah polynomials and some of their applications in the theory of representation of quantum algebras.