On a generalization of Bessel functions satisfying higher-order differential equations

Abstract

Bessel-type functions { J α, M λ ( x)} λ⩾0 with two parameters α ⩾ − 1 2 and M ⩾ 0, which include the classical normalized Bessel function for M = 0, are introduced as a certain confluent limit of Koornwinder's Laguerre-type polynomials {L α,N n(x)} nϵ N 0 . For any α ϵ N 0, they arise as solutions of a spectral λ-dependent differential equation of order 2 α + 4. A necessary and sufficient condition to transform the differential equation into symmetric form is given in terms of an overdetermined system of linear equations. It is shown that for α = 0, 1, 2, a solution of this problem exists and leads to a (symmetric) fourth-, sixth- and eighth-order differential equation, respectively. For any α ⩾ − 1 2 , we also derive a second-order differential equation, but with coefficients depending nonlinearly on the eigenvalue parameter λ. Finally, two differential expressions of order 2α + 2, α ϵ N 0 , are constructed which map the classical Bessel functions onto their nonclassical counterparts, and vice versa. This result may be used to establish an orthogonality relation for the Bessel-type functions (in a distributional sense) with respect to the distributional weight function (2/ Γ( α + 1)) x 2 α+1 + Mδ( x), δ denoting the point mass centered at the origin.