Hostname: page-component-7c8c6479df-94d59 Total loading time: 0 Render date: 2024-03-27T15:45:05.135Z Has data issue: false hasContentIssue false

Orthogonal polynomials satisfying fourth order differential equations

Published online by Cambridge University Press:  14 November 2011

Allan M. Krall
Affiliation:
McAllister Building, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A.

Synopsis

These 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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bognar, J.. Indefinite inner product spaces (New York: Springer, 1974).CrossRefGoogle Scholar
2Bremermann, H.. Distributions, complex variables, and Fourier transforms (Reading, Mass.: Addison-Wesley, 1965).Google Scholar
3Everitt, W. N.. The Sturm-Liouville problem for fourth-order differential equations. Quart. J. Math. 8 (1957), 146160.Google Scholar
4Everitt, W. N.Fourth order singular differential equations. Math. Ann. 49 (1963), 320340.CrossRefGoogle Scholar
5Everitt, W. N. and Krishna Kumar, V.. On the Titchmarsh-Weyl theory of ordinary symmetric differential expressions I: The general theory. Nieuw Arch. Wisk. 24 (1976), 148.Google Scholar
6Everitt, W. N.On the Titchmarsh-Weyl theory of ordinary symmetric differential expressions II: The odd order case. Nieuw Arch. Wisk. 24 (1976), 109145.Google Scholar
7Feller, W.. An introduction to probability theory and its applications, vol. II (New York: Wiley, 1966).Google Scholar
8Fulton, C.. Two point boundary value problems with the eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 293308.CrossRefGoogle Scholar
9Fulton, C. Two point boundary value problems of limit circle type, manuscript.Google Scholar
10Gelfand, I. M. and Shilov, G. E.. Generalized functions, vol. 1 (New York: Academic, 1964).Google Scholar
11Grosswald, E.. Bessel polynomials. Lecture Notes in Mathematics 698 (Berlin: Springer, 1978).Google Scholar
12Krall, A. M.. Adjoint systems in inner product spaces. SIAM J. Appl. Math. 20 (1971), 4043.CrossRefGoogle Scholar
13Krall, A. M.Orthogonal polynomials through moment generating functionals. SIAM J. Math. Anal. 9 (1978), 600603.CrossRefGoogle Scholar
14Krall, H. L.. Certain differential equations for Tchebycheff polynomials. Duke Math. J. 4 (1938), 705718.CrossRefGoogle Scholar
15Krall, H. L. On orthogonal polynomials satisfying a certain fourth order differential equation. The Pennsylvania State College Studies, No. 6. The Pennsylvania State College, State College, Pa., 1940.Google Scholar
16Law, A. G. and Sledd, M. B.. Normalizing orthogonal polynomials by using their recurrence coefficients. Proc. Amer. Math. Soc. 48 (1975), 505507.Google Scholar
17Morton, R. D. and Krall, A. M.. Distributional weight functions for orthogonal polynomials. SIAM J. Math. Anal. 9 (1978), 604626.CrossRefGoogle Scholar
18Rainville, E. D.. Special functions (New York: Macmillan, 1960).Google Scholar
19Shore, S. D.. On the second order differential equation which has orthogonal polynomial solutions. Bull. Calcutta Math. Soc. 56 (1964). 195198.Google Scholar
20Zygmund, A.. Trigonometric series (Warszawa-Lwów, (1935).Google Scholar