On relations of the $T$-matrices arising in the axisymmetric problem of light scattering by a spheroid

The relations between the $T$-matrices emerging when solving the problem of light scattering by a spheroid by applying the expansions of the electro-magnetic fields in the employing spheroidal and spherical bases are found. The behavior of the obtained relations is numerically studied, and it is noted that in a wide range of the task parameter values the calculation of the spheroidal $T$-matrix and its corresponding transformation is the only practical way to derive the spherical $T$-matrix often used in applications.