Higher-order Bessel equations integrable in elementary functions

The eigenfunction problem for a scalar Euler operator leads to an ordinary differential equation, which is an analog of higher-order Bessel equations. Its solutions are expressed through elementary functions in the case where the corresponding Euler operator can be factorized in a certain appropriate way. We obtain a formula describing such solutions. We consider the problem on common eigenfunctions of two Euler operators and present commuting Euler operators of orders $4$, $6$, and $10$ and a formula for their common eigenfunction and also commuting operators of orders $6$ and $9$.