Orthogonal Polynomials With Weight Function (1 - x)α( l + x)β + Mδ(x + 1) + Nδ(x - 1)

1984 ◽  
Vol 27 (2) ◽  
pp. 205-214 ◽  
Author(s):  
Tom H. Koornwinder
Keyword(s):  
Differential Equation ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Hypergeometric Functions ◽  
Order Differential Equation ◽  
Second Order Differential Equations ◽  
Special Cases ◽  
Delta Functions ◽  
Fourth Order Differential Equation

AbstractWe 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 4F3 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.

scholarly journals Orthogonal polynomials associated to a certain fourth order differential equation

2011 ◽  
Vol 26 (3) ◽  
pp. 295-310 ◽  
Author(s):  
Joachim Hilgert ◽  
Toshiyuki Kobayashi ◽  
Gen Mano ◽  
Jan Möllers
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Fourth Order Differential Equation

Orthogonal polynomials satisfying fourth order differential equations

1981 ◽  
Vol 87 (3-4) ◽  
pp. 271-288 ◽  
Author(s):  
Allan M. Krall
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Jacobi Polynomials ◽  
Weight Functions ◽  
Classical Orthogonal Polynomials ◽  
Absolutely Continuous ◽  
Independent Variable ◽  
Fourth Order Differential Equation

SynopsisThese polynomials, which are intimately connected with the Legendre, Laguerre and Jacobi polynomials, are orthogonal with respect to Stieltjes weight functions which are absolutely continuous on (− 1, 1), (0, ∞) and (0, 1), respectively, but which have jumps at some of the intervals' ends. Each set satisfies a fourth order differential equation of the form Ly = λny, where the coefficients of the operator L depends only upon the independent variable. The polynomials also have other properties, which are usually associated with the classical orthogonal polynomials.

scholarly journals Asymptotic theory for a critical case for a general fourth-order differential equation

1998 ◽  
Vol 21 (3) ◽  
pp. 479-488
Author(s):  
A. S. A. Al-Hammadi
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Critical Case ◽  
Asymptotic Theory ◽  
Order Differential Equation ◽  
Fourth Order Equations ◽  
Fourth Order Differential Equation

In this paper we identify a relation between the coefficients that represents a critical case for general fourth-order equations. We obtained the forms of solutions under this critical case.

25.—On the Eigenfunction Expansion associated with a Singular Complex-valued Fourth-order Differential Equation

1976 ◽  
Vol 75 (4) ◽  
pp. 325-332
Author(s):  
Jyoti Chaudhuri ◽  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Representation Theorem ◽  
Eigenfunction Expansion ◽  
Order Differential Equation ◽  
Convergence Theory ◽  
Infinite Intervals ◽  
Complex Valued ◽  
Class Of Functions ◽  
Fourth Order Differential Equation

SynopsisThe direct convergence theory of eigenfunction expansions associated with boundry value problems, not necessarily self-adjoint, generated from complex-valued fourth-order symmetric ordinary differential expressions on semi-infinite intervals, is discussed. An admissible class of functions for the expansion is characterised. Also a generalisation of Stieltjes representation theorem for analytic functions discussed in [13, §§ 22.23 and 24] is obtained.

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