CONNECTION FORMULAS AND REPRESENTATIONS OF LAGUERRE POLYNOMIALS IN TERMS OF THE ACTION OF LINEAR DIFFERENTIAL OPERATORS

Abstract

In this paper, we introduce the notion of Oε-classical orthogonal polynomials, where Oε := I + εD (ε 6= 0). It is shown that the scaled Laguerre polynomial sequence {a −nL (α) n (ax)}n>0, where a = −ε −1 , is actually the only Oε-classical sequence. As an illustration, we deal with some representations of Laguerre polynomials L (0) n (x) in terms of the action of linear differential operators on the Laguerre polynomials L (m) n (x). The inverse connection problem of expanding Laguerre polynomials L (m) n (x) in terms of L (0) n (x) is also considered. Furthermore, some connection formulas between the monomial basis {x n}n>0 and the shifted Laguerre basis {L (m) n (x + 1)}n>0 are deduced.

Keywords

• classical polynomials
• laguerre polynomials
• lowering and raising operators
• structure relations
• higher order differential operators
• connection formulas