Abstract: | We develop a theory of “special functions” associated with a certain fourth-order differential operator Dm,n\mathcal{D}_{\mu,\nu} on ℝ depending on two parameters μ,ν. For integers μ,ν≥−1 with μ+ν∈2ℕ0, this operator extends to a self-adjoint operator on L
2(ℝ+,x
μ+ν+1 dx) with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive
basic properties of the eigenfunctions such as orthogonality, completeness, L
2-norms, integral representations, and various recurrence relations. |