On multiplicity bounds for eigenvalues of the clamped round plate

We ask whether the only multiplicities in the spectrum of the clamped round plate are trivial, i.e., whether all existing multiplicities are due to the isometries of the sphere, or, equivalently, whether any eigenfunction is separated. We prove that any eigenfunction can be expressed as a sum of at most two separated ones, by showing that otherwise the corresponding eigenvalue is algebraic, contradicting the Siegel-Shidlovskii theory. In two dimensions it follows that no eigenvalue is of multiplicity greater than four. The proof exploits a linear recursion of order two for cross-product Bessel functions with coefficients which are not even algebraic functions, though they do satisfy a non-linear algebraic recursion.