FoCM 2014 conference

Workshop C5 - Special Functions and Orthogonal Polynomials

December 20, 14:30 ~ 15:00 - Room A21

## Applications of infinite matrices in the theories of orthogonal polynomials and operational calculus

### Luis Verde-Star

### Universidad Autonoma Metropolitana, Mexico - verde@xanum.uam.mx

We use some algebras of infinite matrices $[a_{j,k}]$, where the indexes run over all the integers, to study sequences of polynomials and formal power series and also for the construction of a general operational calculus that can be used to solve linear functional equations of several types.

We consider infinite matrices of the form $\sum_k D_k X^k$, where the $D_k$ are diagonal matrices, $X$ is a shift, and $D_k \ne 0$ for only a finite number of negative values of $k$. Several basic properties and characterizations of orthogonal polynomial sequences are expressed in terms of infinite matrices.

This work extends some of the results obtained in our previous papers

L. Verde-Star, Characterization and construction of classical orthogonal polynomials using a matrix approach, Linear Algebra Appl. 438 (2013) 3635--3648.

G. Bengochea, L. Verde-Star, Linear algebraic foundations of the operational calculi, Adv. Appl. Math. 47 (2011) 330--351.