The Fourth-order Bessel–type Differential Equation

doi:10.1080/00036810310001599446

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The Fourth-order Bessel–type Differential Equation

Journal Article•DOI•

The Fourth-order Bessel–type Differential Equation

Jyoti Das¹, William Norrie Everitt², Don B. Hinton³, Lance L. Littlejohn⁴, Clemens Markett - Show less +1 more•Institutions (4)

University of Calcutta¹, University of Birmingham², University of Tennessee³, Utah State University⁴

01 Apr 2004-Applicable Analysis (Taylor & Francis Group)-Vol. 83, Iss: 4, pp 325-362

TL;DR: 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.

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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...

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Journal Article•DOI•

Theory of Functions of a Complex Variable

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W. Burnside

01 Jun 1893-Nature

TL;DR: The theory of functions is the basis on which the whole of pure mathematics which deals with continuously varying quantity rests as mentioned in this paper, and the answer would not be too wide nor would it always imply too much.

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Abstract: WHAT is the theory of functions about? This question may be heard now and again from a mathematical student; and if, by way of a pattial reply, it be said that the elements of the theory of functions forms the basis on which the whole of that part of pure mathematics which deals with continuously varying quantity rests, the answer would not be too wide nor would it always imply too much. Theory of Functions of a Complex Variable. By Dr. A. R. Forsyth. (Cambridge University Press, 1893.)

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491 citations

Journal Article•DOI•

Spectral analysis of fourth order differential operators III

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Horst Behncke¹•Institutions (1)

University of Osnabrück¹

01 Jan 2006-Mathematische Nachrichten

TL;DR: In this article, the spectral theory of differential operators of the form============★★★★★★★★★★ Ⴗℒ2w(0, ∞) was studied and estimates for the eigenfunctions and M -matrix were derived.

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Abstract: We study the spectral theory of differential operators of the form on ℒ2w(0, ∞). By means of asymptotic integration, estimates for the eigenfunctions andM -matrix are derived. Since the M -function is the Stieltjes transform of the spectral measure, spectral properties of τ are directly related to the asymptotics of the eigenfunctions. The method of asymptotic integration, however, excludes coefficients which are too oscillatory or whose derivatives decay too slowly. Consequently there is no singular continuous spectrum in all our cases. This was found earlier for Sturm–Liouville operators, for which theWKB method provides a good approximation. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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16 citations

Journal Article•DOI•

Properties of the solutions of the fourth-order Bessel-type differential equation

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William Norrie Everitt¹, Clemens Markett², Lance L. Littlejohn³•Institutions (3)

University of Birmingham¹, RWTH Aachen University², Baylor University³

01 Nov 2009-Journal of Mathematical Analysis and Applications

TL;DR: 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, which leads to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin.

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12 citations

Cites background or methods from "The Fourth-order Bessel–type Differ..."

...In a series of papers [2], [5], [6], [8] and [9] we investigated the spectral theoretical aspects of the fourth-order equation (1....
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...The Bessel-type function J was introduced in [7], and the solution Y 0;M was dened in [2]....
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Posted Content•

Fourth-order Bessel-type special functions: a survey

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W. N. Everitt

01 Sep 2006-arXiv: Classical Analysis and ODEs

TL;DR: In this paper, the authors report on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform.

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Abstract: This survey paper reports on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform. These results are based upon the properties of the classical Bessel and Laguerre secondorder differential equations, and on the fourth-order Laguerre-type differential equation. From these differential equations and their solutions, limit processes yield the fourth-order Bessel-type functions and the associated differential equation.

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10 citations

Posted Content•

The Fourth-Order Type Linear Ordinary Differential Equations

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W. N. Everitt¹, D. J. Smith¹, M. van Hoeij²•Institutions (2)

University of Birmingham¹, Florida State University²

21 Mar 2006-arXiv: Classical Analysis and ODEs

TL;DR: In this article, the authors report on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type and Legendre-type.

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Abstract: This note reports on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type, Legendre-type.

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10 citations

Cites background or methods from "The Fourth-order Bessel–type Differ..."

...W.N. EVERITT, D. J. SMITH, AND M. VAN HOEIJ For the Bessel-type equation two independent solution were obtained, see [8] and [3], and then a complete set of four independent solutions in the papers [14] and [4]; these solutions are dependent upon the classical Bessel functions J r , Y r , I r , K r…...
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...The properties of the fourth-order Bessel-type functions have been studied in the papers [3], [5] and [6]....
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References

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Journal Article•DOI•

On the transformation theory of ordinary second-order linear symmetric differential expressions

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W. N. Everitt

01 Jan 1982-Czechoslovak Mathematical Journal

35 citations

"The Fourth-order Bessel–type Differ..." refers background in this paper

...6) gives an isomorphic isometric mapping, see [11], between the spaces L(2)ðð0,1Þ; xÞ and L(2)ð0,1Þ, under which mapping there is a unitary equivalence between associated self-adjoint operators generated by the respective differential expressions, in these two spaces....
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Journal Article•DOI•

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

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William Norrie Everitt¹, C. Markett²•Institutions (2)

University of Birmingham¹, RWTH Aachen University²

20 Oct 1994-Journal of Computational and Applied Mathematics

TL;DR: In this paper, a necessary and sufficient condition to transform the Bessel function into a symmetric form is given in terms of an overdetermined system of linear equations, which leads to a (symmetric) fourth-, sixth-and eighth-order differential equation, respectively.

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32 citations

"The Fourth-order Bessel–type Differ..." refers background or methods or result in this paper

...1), for all 2 C, and hence for all 2 C and all M > 0: Similar arguments to the methods given in [16] show that the function defined by...
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...Information about the higher even-order Bessel-type differential equations is given in [16]; in particular the explicit Lagrange symmetric forms of the sixth-order and THE FOURTH-ORDER BESSEL-TYPE 327...
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...This notation has been adopted as is introduced into the analysis of [16] for other than spectral purposes....
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...1) in view of the use of the symbol in [16] for purposes involving the theory of special functions....
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Journal Article•DOI•

The Spectrum of Differential Operators of Order 2n with Almost Constant Coefficients

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Horst Behncke¹, Don B. Hinton², Christian Remling¹•Institutions (2)

University of Osnabrück¹, University of Tennessee²

01 Sep 2001-Journal of Differential Equations

TL;DR: In this paper, the spectral properties of higher-order ODEs were discussed and uniform asymptotic integration techniques were used to analyze the associated differential equations with the help of uniform integration techniques.

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30 citations

Book Chapter•DOI•

The Glazman-Krein-Naimark theorem for ordinary differential operators

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W. N. Everitt¹, L. Markus²•Institutions (2)

University of Birmingham¹, University of Minnesota²

01 Nov 1997

TL;DR: The original form of symmetric boundary conditions for Lagrange symmetric (formally self-adjoint) ordinary linear differential expressions was obtained by I.M. Glazman in his seminal paper of 1950 as mentioned in this paper.

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Abstract: The original form of symmetric boundary conditions for Lagrange symmetric (formally self-adjoint) ordinary linear differential expressions, was obtained by I.M. Glazman in his seminal paper of 1950. This result described all self-adjoint differential operators, in the underlying Hilbert function space, generated by real even-order differential expressions.

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23 citations

"The Fourth-order Bessel–type Differ..." refers background or methods in this paper

...1) is regular at each point of ð0,1Þ; the equation is singular at 0þ and at þ1: For these notations see the book by Naimark [25, Chapter V], and the paper by Everitt and Markus [17]....
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...SELF-ADJOINT OPERATORS IN L2ðð0,?Þ; x Þ The self-adjoint extensions of the closed symmetric operator T0 are determined by the GKN-theorem on boundary conditions as given in [25, Chapter V] and in [17]....
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...BASIC DIFFERENTIAL OPERATORS IN L2ðð0,?Þ; x Þ The maximal and the minimal differential operators, denoted respectively T1 and T0, generated by the differential expression LM in the Hilbert function space L(2)ðð0,1Þ; xÞ are defined as follows, see [17] and [25, Chapter V, Section 17]: (i) T1 : DðT1Þ L(2)ðð0,1Þ; xÞ ! L(2)ðð0,1Þ; xÞ by DðT1Þ :1⁄4 f f 2 DðLMÞ: f , x (1)LMð f Þ 2 L(2)ðð0,1Þ; xÞg ð5:1Þ THE FOURTH-ORDER BESSEL-TYPE 333...
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...On the interval ð0,1Þ R and considered in the space L1locð0,1Þ, the Lagrange symmetric differential equation (3.1) is regular at each point of ð0,1Þ; the equation is singular at 0þ and at þ1: For these notations see the book by Naimark [25, Chapter V], and the paper by Everitt and Markus [17]....
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Journal Article•

Differential operators and the Legendre type polynomials

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William Norrie Everitt, Lance L. Littlejohn

01 Jan 1988-Differential and Integral Equations

22 citations

"The Fourth-order Bessel–type Differ..." refers methods in this paper

...Previous studies of fourth-order differential equations generating the Legendre-type, Jacobi-type and Laguerre-type orthogonal polynomials, see [12,13,15], have shown that an initial study of the spectral properties of the differential equation in the classical Hilbert function space is essential to the subsequent study of spectral properties in the jump-weighted Hilbert space....
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