New development in theory of Laguerre polynomials

TL;DR

This paper introduces new complete orthonormal sets of -Laguerre type polynomials derived from a Schrödinger equation framework, with applications to atomic integrals and potential expansions, emphasizing their completeness and convergence properties.

Contribution

The paper presents a novel set of -Laguerre type polynomials based on a Schrödinger equation approach, expanding their applications in atomic physics and potential theory.

Findings

New -LTP are complete without continuum states.

Standard and nonstandard -LTP conventions are equivalent.

Derived expansion formulas for atomic integrals and tested convergence.

Abstract

The new complete orthonormal sets of -Laguerre type polynomials (-LTP,) are suggested. Using Schr\"odinger equation for complete orthonormal sets of -exponential type orbitals (-ETO) introduced by the author, it is shown that the origin of these polynomials is the centrally symmetric potential which contains the core attraction potential and the quantum frictional potential of the field produced by the particle itself. The quantum frictional forces are the analog of radiation damping or frictional forces suggested by Lorentz in classical electrodynamics. The new -LTP are complete without the inclusion of the continuum states of hydrogen like atoms. It is shown that the nonstandard and standard conventions of -LTP and their weight functions are the same. As an application, the sets of infinite expansion formulas in terms of -LTP and L-Generalized Laguerre polynomials (L-GLP) for atomic nuclear attraction integrals of Slater type orbitals (STO) and Coulomb-Yukawa like correlated interaction potentials (CIP) with integer and noninteger indices are obtained. The arrange and rearranged power series of a general power function are also investigated. The convergence of these series is tested by calculating concrete cases for arbitrary values of parameters of orbitals and power function.