On the eigenfunctions corresponding to the bandpass kernel, in the case of degeneracy

It has previously been pointed out that the eigenfunctions of the finite integral equation with bandlimited difference kernel satisfy a certain second order linear differential equation, containing one parameter, whose continuous solutions, for discrete values of the parameter, are the prolate spheroidal wave functions. We consider here the finite integral equation with bandpass difference kernel. It is shown that, in the case of degeneracy, one eigenfunction is the continuous solution of a certain fourth order linear differential equation, containing two parameters which must be determined from prescribed conditions. The second eigenfunction is the derivative of the first one.