Representation of doubly infinite matrices as non-commutative Laurent series

Abstract

We present a new way to deal with doubly in nite lower Hessenberg matrices based on the representation of the matrices as the sum of their diagonal submatrices. We show that such representation is a simple and useful tool for computation purposes and also to obtain general properties of the matrices related with inversion, similarity, commutativity, and Pincherle derivatives. The diagonal representation allows us to consider the ring of doubly in nite lower Hessenberg matrices over a ring R as a ring of Laurent series in one indeterminate, with coe cients in the ring of R-valued sequences that don't commute with the indeterminate.

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

“…A more detailed account of the matrix approach to polynomial sequences can be found in [14] and [15]. See also [1] where some properties of doublyinfinite matrices are obtained.…”

Maximal and Minimal Triangular Matrices

“…that is not a polynomial of j ∞ (0). Nevertheless, it should be mentioned that all the elements of C T∞(F ) (j ∞ (0)) are simply the upper triangular Toeplitz matrices that can be written as the formal power series ∞ i=0 α i j i ∞ (0) (see [1], but also [2]). Moreover, the centralizer of the algebra of upper triangular Toeplitz matrices is the same algebra, so we have…”

Estudio de familias de sucesiones de polinomios ortogonales utilizando matrices infinitas

“…En el Capítulo 5 se extiende el estudio de las matrices infinitas al estudio de matrices doblemente infinitas, donde los índices de los elementos de éstas corren sobre el conjunto de los números enteros, los resultados alcanzados aparecen en [5]. Se comienza presentando el conjunto de matrices doblemente infinitas, L, y se mencionan propiedades y características.…”

“…El objetivo de este Capítulo es extender el reciente desarrollo sobre la teoría de matrices infinitas a matrices doblemente infinitas, presentada en [5]. Esto se traduce en que los problemas analíticos que aparecen en la multiplicación de matrices infinitas, y la continuidad de los operadores representados por tales matrices se estudiaron ampliamente, pero el estudio de las propiedades puramente algebraicas de las matrices infinitas no ha recibido la atención que se merece.…”