New relationships relating acoustical and electromagnetic beam shape coefficients

It has been recently demonstrated that the electromagnetic beam shape coefficients $g_{n,X}^m$ ($X=TM$ or $TE$) which encode the structure of structured light beams may be expressed in terms of scalar, more specifically acoustical, beam shape coefficients. Because the technique used to obtain the relevant expressions relied on the properties of what is known as the finite series method, the aforementioned expressions were different, depending on whether $(n-m)$ is even or odd. For a reason discussed in the bulk of the paper, it became obvious that the expressions obtained for different parities of $(n-m)$ could be unified. Proceeding to such an unification, the expressions previously published were not only unified, but furthermore simplified, then allowing for an easier and less time-consuming numerical implementation.