The Fourth-order Bessel–type Differential Equation
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.
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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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.
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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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.
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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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.
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 de ned in [2]....
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Posted Content•
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.
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.
10 citations
Posted Content•
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.
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.
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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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.
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.)
491 citations
TL;DR: In this paper, the authors studied orthogonal polynomials for which the weight function is a linear combination of the Jacobi weight function and two delta functions at 1 and 1.
Abstract: We study orthogonal polynomials for which the weight function is a linear combination of the Jacobi weight function and two delta functions at 1 and — 1. These polynomials can be expressed as 4 F3 hypergeometric functions and they satisfy second order differential equations. They include Krall's Jacobi type polynomials as special cases. The fourth order differential equation for the latter polynomials is derived in a more simple way. 0. Introduction. The nonclassical orthogonal polynomials which are eigen- functions of a fourth order differential operator were classified by H. L. Krall (6), (7). These polynomials were described in more details by A. M. Krall (5). The corresponding weight functions are special cases of the classical weight functions together with a delta function at the end point(s) of the interval of orthogonality. A number of A. M. Krall's results can be obtained in a more satisfactory way: (a) Jacobi, Legendre and Laguerre type polynomials are connected with each other by quadratic transformations and a limit formula. (b) The power series for the Jacobi type polynomials is of 3F2-type. (c) There is a pair of second order differential operators not depending on n which connect the Jacobi polynomials P£*'0)(2x -1) and the Jacobi type polyno mials Sn(x). Combination of these two differentiation formulas yields the fourth order equation for Sn(x). It is the first purpose of the present paper to make these comments to (5). The second purpose is to describe a more general class of Jacobi type polynomials, with weight function (l-x) a (l + ;x)3 + linear combination of 8(x +1) and 8(x — 1). They can be expressed in terms of Jacobi polynomials as ((anx + bn)d/dx + cn)P^'3)(x) for certain coefficients a", bn, cn and their power series in |(l-x) is of 4 F3 type. Finally, they satisfy a second order differential equation with polynomial coefficients depending on n, but of bounded degree, thus generalizing the known result for the Jacobi type polynomials Sn(x) (cf. Littlejohn & Shore (9)) and providing further examples for the general theory
191 citations
Book•
01 Jan 1982
128 citations
"The Fourth-order Bessel–type Differ..." refers background in this paper
...For a general discussion on results for differential operators under the general title of virial theorems see the book [10]....
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01 Apr 1991
TL;DR: In this paper, it was shown that the generalized Laguerre polynomials (a+ for N > 0) satisfy a unique differential equation of the form 00 N ai (x)y(1)(x) + xy "(x)+ ( + 1-x)-y'(x)+ ny(x)) = 0, 1=0 where {a,(x)}1o are continuous functions on the real line and {a'(ex)1i are independent of the degree n.
Abstract: Koornwinder's generalized Laguerre polynomials {L` N(X)}oo0 are orthogonal on the interval [0, oo) with respect to the weight function 1 f-)xae x + N3(x), (x > -1, N > 0. We show that these polynomials ['(a+ for N > 0 satisfy a unique differential equation of the form 00 N ai (x)y(1)(x) + xy " (x) + ( + 1-x)y'(x) + ny(x) = 0, 1=0 where {a,(x)}1o are continuous functions on the real line and {a'(x)}1i are independent of the degree n . If N > 0, only in the case of nonnegative integer values of a this differential equation is of finite order.
106 citations