Shift operators and recurrence relations for individual Lamé polynomials

Abstract

Our contribution to Gr31 “Review of the development and application of spheroconal theory of angular momentum” included a mention of our current work on individual Lamé polynomials. This written version reports original results on shift operators and recurrence relations connecting a Lamé polynomial with angular momentum l, species [A] and excitation n, with neighbouring polynomials with $$EQUATION$$l’ = l±1, species [A’] where the prime indicates a derivative, and excitations n’ in their different possibilities, for the derivative operator. Other operators involve multiplication by another Lamé polynomial, starting with the singularity removing factors Ai = 1, sn, cn, dn, sncn, sndn, cndn, sncndn as monomials. The successive and complementary use of these operators for Lamé polynomials in their two respective degrees of freedom connects with the ladder and shift operator actions of cartesian components of the angular momentum and linear momentum on the product of those polynomials as rotational eigenstates. The identification of the operators for individual Lamé polynomials fills a gap in the study of their properties and connections.