Journal ArticleDOI
The Fourth-order Bessel–type Differential Equation
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TLDR
The Bessel-type functions as discussed by the authors are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity on the point of infinity of the complex plane, which are derived by linear combinations and limit processes from the classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials.Abstract:
The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane. There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation. When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coeffic...read more
Citations
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Journal ArticleDOI
Fourth-order Bessel equation: eigenpackets and a generalized Hankel transform †
TL;DR: In this article, the authors proved that S M has a complete eigenpacket, which is reflected in a distributional orthogonality on which the expansion theorems are based.
Journal ArticleDOI
Additional spectral properties of the fourth-order Bessel-type differential equation
TL;DR: The spectral properties of the self-adjoint differential operator generated by the fourth-order Bessel-type differential expression, as defined by Everitt and Markett in 1994, in a Lebesgue-Stieltjes Hilbert function space are discussed in this paper.
Journal ArticleDOI
Quasi-separation of the biharmonic partial differential equation
TL;DR: In this article, the authors considered analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane.
Journal ArticleDOI
New representation and factorizations of the higher-order ultraspherical-type differential equations
TL;DR: In this article, the orthogonality relation of the ultraspherical-type polynomials was deduced directly from the differential equation property, and two types of factorizations of the corresponding differential operators of order 2 α + 4 into a product of α + 2 linear second-order operators were introduced.
Journal ArticleDOI
Towards a generalization of the separation of variables technique
TL;DR: In this article, a generalized form of the method of reduction for higher-order linear partial differential equations incorporating mixed derivatives is presented. But it is only applied to a limited number of differential operators both linear and non-linear.
References
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