On a generalization of Bessel functions satisfying higher-order differential equations
The aim of the authors is to introduce a generalization of the Bessel functions, which satisfy a higher-order differential equation and give rise to a Hankel transform. The generalization is realized by a confluent limit process applied to Laguerre-type polynomials \(L_n^{\alpha, N} (x)\). The new functions (Bessel-type functions) which are shown by \(J_\lambda^{\alpha, M} (x)\) with \(\alpha\geq -1/2\), \(M\geq 0\) and \(\lambda\geq 0\) satisfy a differential equation of order \(2\alpha +4\) which reduces to the classical Bessel equation when \(M=0\). A necessary and sufficient condition (through \(2\alpha+4\) linear equations with \(\alpha+2\) unknowns) to transform the generalized equation into a formally symmetric one is also given. It is also shown that the functions \(J_\lambda^{\alpha, M} (x)\) have an orthogonality relation with respect to the distributional weight function \((2/ \Gamma (\alpha+ 1))x^{2 \alpha+1}+ M\delta (x)\), namely: \[ {2\over {\Gamma (\alpha+ 1)}} \int_0^\infty J_\lambda^{\alpha, M} (x) J_\mu^{\alpha, M} (x) x^{2\alpha +1} dx+ MJ_\lambda^{\alpha, M} (0) J_\mu^{\alpha, M} (0)= A\delta (\lambda- \mu), \] where \(\delta (x)\) stands for the Dirac's delta function while \(A\) is a known coefficient depending on \(\alpha\), \(\lambda\) and \(M\).
scientific article; zbMATH DE number 739563
scientific article