Some relations between the spheroidal and spherical wave functions

We find new relations between the spheroidal and spherical wave functions as well as between the spheroidal functions related to different spheroidal coordinate systems. The systems should have a common origin of coordinate and a common symmetry axis of coordinate surfaces. The applicability ranges of the relations obtained are discussed. Numerical test calculations have demonstrated the high efficiency of the relations in particular those for wave functions including the radial functions of the first kind. As a particular case we consider the relations between the prolate and oblate spheroidal wave functions including the radial functions of the first and second kinds. These relations are necessary to solve the light scattering problem for nonconfocal layered spheroidal particles.