Generalized Adler–Moser and Loutsenko Polynomials for Point Vortex Equilibria

Equilibrium configurations of point vortices with circulations of two discrete values are associated with the zeros of a sequence of polynomials having many continuous parameters: the Adler–Moser polynomials in the case of circulation ratio −1, and the Loutsenko polynomials in the case of ratio −2. In this paper a new set of polynomial sequences generalizing the vortex system to three values of circulations is constructed. These polynomials are extensions of the previously known polynomials in the sense that they are special cases of the latter when the third circulation is zero. The polynomials are naturally connected with rational functions that satisfy a second-order differential equation.