Fourth-order Bessel equation: eigenpackets and a generalized Hankel transform

In connection with the fourth-order Bessel-type differential equation (L(M)y)(x):= (xy") (x))" - ((9x(-1) + 8M(-1)x)y'(x))' = Lambda xy(x) (x > 0) two expansion theorems are established, the convergence being pointwise or in an L-2-setting. If the positive parameter M tends to zero, these two expansion theorems reduce to the classical Hankel transform of order zero. In a previous article, the authors have proved that in one of the introduced Lebesgue - Stieltjes Hilbert function spaces, the differential expression x(-1) L-M gives rise to exactly one self-adjoint operator S-M. In this article, it is proved, together with the corresponding expansion theorems, that S-M has a complete eigenpacket. The orthogonality property of this eigenpacket is reflected in a distributional orthogonality on which the expansion theorems are based.