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Clemens Markett - One of the best experts on this subject based on the ideXlab platform.

  • properties of the solutions of the fourth order bessel type differential equation
    Journal of Mathematical Analysis and Applications, 2009
    Co-Authors: W N Everitt, Clemens Markett, L L Littlejohn
    Abstract:

    The structured Bessel-type functions of arbitrary even-order were introduced by Everitt and Markett in 1994; these functions satisfy linear ordinary differential equations of the same even-order. The differential equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity. They are all natural extensions of the classical second-order Bessel differential equation. Further these differential equations have real-valued coefficients on the positive real half-line of the plane, and can be written in Lagrange symmetric (Formally self-Adjoint) Form. In the fourth-order case, the Lagrange symmetric differential expression generates self-Adjoint unbounded operators in certain Hilbert function spaces. These results are recorded in many of the papers here given as references. It is shown in the original paper of 1994 that in this fourth-order case one solution exists which can be represented in terms of the classical Bessel functions of order 0 and 1. The existence of this solution, further aided by computer programs in Maple, led to the existence of a linearly independent basis of solutions of the differential equation. In this paper a new proof of the existence of this solution base is given, on using the advanced theory of special functions in the complex plane. The methods lead to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane.

  • the fourth order bessel type differential equation
    Applicable Analysis, 2004
    Co-Authors: Jyoti Das, W N Everitt, D B Hinton, L L Littlejohn, Clemens Markett
    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...

L L Littlejohn - One of the best experts on this subject based on the ideXlab platform.

  • properties of the solutions of the fourth order bessel type differential equation
    Journal of Mathematical Analysis and Applications, 2009
    Co-Authors: W N Everitt, Clemens Markett, L L Littlejohn
    Abstract:

    The structured Bessel-type functions of arbitrary even-order were introduced by Everitt and Markett in 1994; these functions satisfy linear ordinary differential equations of the same even-order. The differential equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity. They are all natural extensions of the classical second-order Bessel differential equation. Further these differential equations have real-valued coefficients on the positive real half-line of the plane, and can be written in Lagrange symmetric (Formally self-Adjoint) Form. In the fourth-order case, the Lagrange symmetric differential expression generates self-Adjoint unbounded operators in certain Hilbert function spaces. These results are recorded in many of the papers here given as references. It is shown in the original paper of 1994 that in this fourth-order case one solution exists which can be represented in terms of the classical Bessel functions of order 0 and 1. The existence of this solution, further aided by computer programs in Maple, led to the existence of a linearly independent basis of solutions of the differential equation. In this paper a new proof of the existence of this solution base is given, on using the advanced theory of special functions in the complex plane. The methods lead to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane.

  • the fourth order bessel type differential equation
    Applicable Analysis, 2004
    Co-Authors: Jyoti Das, W N Everitt, D B Hinton, L L Littlejohn, Clemens Markett
    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...

W N Everitt - One of the best experts on this subject based on the ideXlab platform.

  • properties of the solutions of the fourth order bessel type differential equation
    Journal of Mathematical Analysis and Applications, 2009
    Co-Authors: W N Everitt, Clemens Markett, L L Littlejohn
    Abstract:

    The structured Bessel-type functions of arbitrary even-order were introduced by Everitt and Markett in 1994; these functions satisfy linear ordinary differential equations of the same even-order. The differential equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity. They are all natural extensions of the classical second-order Bessel differential equation. Further these differential equations have real-valued coefficients on the positive real half-line of the plane, and can be written in Lagrange symmetric (Formally self-Adjoint) Form. In the fourth-order case, the Lagrange symmetric differential expression generates self-Adjoint unbounded operators in certain Hilbert function spaces. These results are recorded in many of the papers here given as references. It is shown in the original paper of 1994 that in this fourth-order case one solution exists which can be represented in terms of the classical Bessel functions of order 0 and 1. The existence of this solution, further aided by computer programs in Maple, led to the existence of a linearly independent basis of solutions of the differential equation. In this paper a new proof of the existence of this solution base is given, on using the advanced theory of special functions in the complex plane. The methods lead to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane.

  • the fourth order bessel type differential equation
    Applicable Analysis, 2004
    Co-Authors: Jyoti Das, W N Everitt, D B Hinton, L L Littlejohn, Clemens Markett
    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...

Jyoti Das - One of the best experts on this subject based on the ideXlab platform.

  • the fourth order bessel type differential equation
    Applicable Analysis, 2004
    Co-Authors: Jyoti Das, W N Everitt, D B Hinton, L L Littlejohn, Clemens Markett
    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...

C S Chang - One of the best experts on this subject based on the ideXlab platform.