Fourier–Bessel-type series: The fourth-order case

Abstract

The structured higher-order Bessel-type linear ordinary differential equations were first discovered in 1994. There is a denumerable infinity of these higher-order equations, all of then of even-order. These differential equations possess many of the properties of the classical second-order Bessel differential equation, but these higher-order cases bring remarkable new analytic structures. In many ways it is sufficient to study the properties of the fourth-order Bessel-type differential equation to be able to assess the corresponding properties of the sixth-and higher-order cases. This paper follows a number of earlier papers devoted to the study of the fourth-order case. These publications show the connections between the special function properties of solutions of the differential equation, and the properties of linear differential operators generated by the associated linear differential expression in certain weighted Lebesgue, and Lebesgue–Stieltjes function spaces. To follow the earlier papers on the study of the fourth-order Bessel-type differential equation, this present paper determines the form of the Fourier–Bessel-type series which best extends the classical theory of the second-order Fourier–Bessel series. In fact the Fourier–Bessel-type series are based on a new orthogonal system in terms of the regular eigensolutions of the fourth-order Bessel-type equation. The corresponding eigenvalues are obtained by restricting the spectral parameter to the zeros of an analytic function arising already in the Dini boundary conditions.