On the Mathematical Nature of Guseinov's Rearranged One-Range Addition Theorems for Slater-Type Functions

TL;DR

This paper critically examines Guseinov's rearranged addition theorems for Slater-type functions, highlighting mathematical issues in transforming Laguerre expansions into pointwise convergent series.

Contribution

It provides a detailed analysis of the mathematical validity of Guseinov's rearrangements, emphasizing the differences between truncated and infinite series expansions.

Findings

Rearranged addition theorems may lack mathematical legitimacy.

Laguerre expansions converge only in the mean, not pointwise.

Transforming to power series can lead to convergence issues.

Abstract

Starting from one-range addition theorems for Slater-type functions, which are expansion in terms of complete and orthonormal functions based on the generalized Laguerre polynomials, Guseinov constructed addition theorems that are expansions in terms of Slater-type functions with a common scaling parameter and integral principal quantum numbers. This was accomplished by expressing the complete and orthonormal Laguerre-type functions as finite linear combinations of Slater-type functions and by rearranging the order of the nested summations. Essentially, this corresponds to the transformation of a Laguerre expansion, which in general only converges in the mean, to a power series, which converges pointwise. Such a transformation is not necessarily legitimate, and this contribution discusses in detail the difference between truncated expansions and the infinite series that result in the absence of truncation