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A Linear Algebra approach to monomiality and operational methods

Luis Verde-Star

Abstract

We use linear algebraic methods to obtain general results about linear operators on a space of polynomials that we apply to the operators associated with a polynomial sequence by the monomiality property. We show that all such operators are differential operators with polynomial coefficients of finite of infinite order. We consider the monomiality operators associated with several classes of polynomial sequences, such as Appell and Sheffer, and also orthogonal polynomial sequences that include the Meixner, Krawtchouk, Laguerre, Meixner-Pollaczek, and Hermite families.

A Linear Algebra approach to monomiality and operational methods

Abstract

We use linear algebraic methods to obtain general results about linear operators on a space of polynomials that we apply to the operators associated with a polynomial sequence by the monomiality property. We show that all such operators are differential operators with polynomial coefficients of finite of infinite order. We consider the monomiality operators associated with several classes of polynomial sequences, such as Appell and Sheffer, and also orthogonal polynomial sequences that include the Meixner, Krawtchouk, Laguerre, Meixner-Pollaczek, and Hermite families.
Paper Structure (12 sections, 6 theorems, 114 equations)

This paper contains 12 sections, 6 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. The algebra of generalized lower Hessenberg matrices
  3. Sequences of polynomials and formal power series
  4. The basic shifts and differentiation operators
  5. Differential operators on the space of polynomials
  6. Monomiality operators of general polynomial sequences
  7. Differential operator representation of matrices of negative index
  8. Monomiality and Appell polynomial sequences
  9. Modified composition matrices and polynomial sequences of binomial type
  10. Sheffer polynomial sequences
  11. Some orthogonal polynomial sequences
  12. Series of generalized derivatives

Key Result

Proposition 2.1

Let $A$ be an element of the group $\mathcal{G}$ and let $U$ be an element of $\mathcal{L}$. Then we have (i) There exists a unique $U_r$ in $\mathcal{L}$ such that $A U_r= U A$. (ii) There exists a unique $U_\ell$ in $\mathcal{L}$ such that $U_{\ell} A = A U$.

Theorems & Definitions (6)

  • Proposition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Theorem 8.1