The Fourth-order Bessel–type Differential Equation

doi:10.1080/00036810310001599446

Home
/
Papers
/
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.

read less

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 moreread less

Citations

PDF

Open Access

More filters

Journal Article•DOI•

Theory of Functions of a Complex Variable

[...]

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.

...read moreread less

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.)

...read moreread less

491 citations

Journal Article•DOI•

Spectral analysis of fourth order differential operators III

[...]

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.

...read moreread less

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)

...read moreread less

16 citations

Journal Article•DOI•

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

[...]

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.

...read moreread less

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....
[...]
...The Bessel-type function J was introduced in [7], and the solution Y 0;M was dened in [2]....
[...]

Posted Content•

Fourth-order Bessel-type special functions: a survey

[...]

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.

...read moreread less

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.

...read moreread less

10 citations

Posted Content•

The Fourth-Order Type Linear Ordinary Differential Equations

[...]

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.

...read moreread less

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.

...read moreread less

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…...
[...]
...The properties of the fourth-order Bessel-type functions have been studied in the papers [3], [5] and [6]....
[...]

1
2
3
4
…

References

PDF

Open Access

More filters

Journal Article•DOI•

Product formulas and convolution structure for Fourier-Bessel series

[...]

Clemens Markett

01 Dec 1989-Constructive Approximation

TL;DR: In this paper, it was shown that the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts, the first part is the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fouriers-Borschtein differential equation.

...read moreread less

Abstract: One of the most far-reaching qualities of an orthogonal system is the presence of an explicit product formula. It can be utilized to establish a convolution structure and hence is essential for the harmonic analysis of the corresponding orthogonal expansion. As yet a convolution structure for Fourier-Bessel series is unknown, maybe in view of the unpractical nature of the corresponding expanding functions called Fourier-Bessel functions. It is shown in this paper that for the half-integral values of the parameter\(\alpha = n + \frac{1}{2}\),n=0, 1, 2,⋯, the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts. While the first part is still the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fourier-Bessel differential equation. It is given, essentially, as a finite sum over triple products of Bessel polynomials. The representation is explicit up to coefficients which are calculated here for the first two nontrivial cases\(\alpha = \frac{3}{2}\) and\(\alpha = \frac{5}{2}\). As a consequence, a positive convolution structure is established for\(\alpha \in \{ \frac{1}{2},\frac{3}{2},\frac{5}{2}\}\). The method of proof is based on solving a hyperbolic initial boundary value problem.

...read moreread less

19 citations

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

...Applicable Analysis Vol. 83, No. 4, April 2004, pp. 325–362 The Fourth-order Bessel-type Differential Equation JYOTI DASa,W.N. EVERITTb,*, D.B.HINTONc, L.L.LITTLEJOHNd and C.MARKETTe aDepartment of Pure Mathematics, University of Calcutta, 35 Bally Gunge Circular Road, Calcutta 700 019, India; bSchool of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT,UK; cDepartment of Mathematics, University of Tennessee, Knoxville,TN 37996,USA; dDepartment of Mathematics and Statistics, Utah State University, Logan, UT 84332-3900, USA; eLehrstuhl A fu« r Mathematik, R-W TH,Templergraben 55, D-52062 Aachen,Germany Communicatedby P. Butzer (Received 6 January 2003; In final form12 March 2003) The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994....
[...]
...This process is best illustrated through the following diagram, see [16, Section 1, p. 328] (for the first two lines of this table see the earlier work of Koornwinder [23] and Markett [24]): Jacobi polynomials kð , Þð1 xÞ ð1þ xÞ !...
[...]
...328] (for the first two lines of this table see the earlier work of Koornwinder [23] and Markett [24]):...
[...]
...The differential equation (1.1) is derived in the paper [16, Section 1, (1.10a)], by Everitt and Markett....
[...]

Journal Article•DOI•

Zeros of combinations of bessel functions and their derivatives

[...]

Mourad E. H. Ismail, Martin E. Muldoon

01 Jan 1988-Applicable Analysis

TL;DR: In this paper, the Mittag-Leffler partial fraction expansion was used to study the monotonicity properties of the zeros of real functions where α, β, τ and δ are real.

...read moreread less

Abstract: We use the Mittag-Leffler partial fractions expansion of and certain monotonicity properties of this and related functions to study the zeros of where α, β, τ and δ are real. Thus we simplify and, in roost cases, strengthen results of E. K Ifantis and P. D. Siafarikas, Applicable Anal. 23 (1986), 85-110. We derive further properties of these zeros in certain special cases

...read moreread less

15 citations

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

...6) are considered by Ismail and Muldoon [20]; in particular there are results on the existence of real zeros of these special functions....
[...]
...Some properties of the combined classical Bessel functions (1.4) and (1.6) are considered by Ismail and Muldoon [20]; in particular there are results on the existence of real zeros of these special functions....
[...]

Journal Article•DOI•

Positivity and discrete spectra for differential operators

[...]

Thomas T. Read¹•Institutions (1)

Western Washington University¹

01 Jan 1982-Journal of Differential Equations

TL;DR: Theorem 5.1.1 as discussed by the authors is a generalization of Theorem 2.1 (see Section 3.1) of Read [1] for two-term expressions in which the bottom coefficient p, > 0.

...read moreread less

15 citations

Journal Article•DOI•

Differential operators and the Laguerre type polynomials

[...]

William Norrie Everitt, Allan M. Krall, Lance L. Littlejohn, V. P. Onyango-Otieno

01 May 1992-Siam Journal on Mathematical Analysis

TL;DR: In this paper, the Laguerre type expression is further studied in the right-definite setting and the appropriate leftdefinite problem associated with the fourth-order Laguerare type differential expression is discussed in detail.

...read moreread less

Abstract: In 1940, all fourth-order differential equations which have a sequence of orthogonal polynomial eigenfunctions were classified by H. L. Krall, up to a linear change of variable. One of these equations was subsequently named the Laguerre type equation and various properties of the orthogonal polynomial solutions and the right-definite boundary value problem were studied by A. M. Krall in 1981. In this paper, the Laguerre type expression is further studied in the right-definite setting and the appropriate left-definite problem associated with the fourth-order Laguerre type differential expression is discussed in detail.

...read moreread less

10 citations

Book Chapter•DOI•

Some Spectral Properties of the Heun Differential Equation

[...]

P. B. Bailey¹, William Norrie Everitt², D. B. Hinton³, Anton Zettl¹•Institutions (3)

Northern Illinois University¹, University of Birmingham², University of Tennessee³

01 Jan 2002

TL;DR: In this paper, a special boundary value problem for the Heun linear ordinary differential equation, derived from a physical problem first studied by the methods of applied mathematics, is studied, which arises from merging singularities in the plastic deformation of crystalline materials under stress.

...read moreread less

Abstract: This paper is concerned with the analytical study of a special boundary value problem for the Heun linear ordinary differential equation, derived from a physical problem first studied by the methods of applied mathematics. This physical problem arises from merging singularities in the plastic deformation of crystalline materials under stress, first studied by Lay and Slavyanov.

...read moreread less

7 citations