Confluent functions, Laguerre polynomials and their (generalized) bilinear integrals

TL;DR

This paper reviews confluent functions and Laguerre polynomials, focusing on their bilinear integrals, and introduces a generalized integral approach that extends the range of computable parameters, revealing orthogonality properties.

Contribution

It provides a comprehensive analysis of bilinear integrals of Laguerre polynomials using generalized integrals, expanding understanding of their orthogonality and pseudo-orthogonality.

Findings

Generalized integrals enable computation beyond traditional convergence limits.

Laguerre polynomials are orthogonal or pseudo-orthogonal depending on parameters.

Orthogonality relations are complex for negative integer parameters.

Abstract

We review properties of confluent functions and the closely related Laguerre polynomials, and determine their bilinear integrals. As is well-known, these integrals are convergent only for a limited range of parameters. However, when one uses the generalized integral they can be computed essentially without restricting the parameters. This gives the (generalized) Gram matrix of Laguerre polynomials. If the parameters are not negative integers, then Laguerre polynomials are orthogonal, or at least pseudo-orthogonal in the case of generalized integrals. For negative integer parameters, the orthogonality relations are more complicated.