A method of constructing Krall's polynomials

✍ Scribed by Alexei Zhedanov

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 142 KB
• Volume: 107
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(99)00070-9

✦ Synopsis

We propose a method of constructing orthogonal polynomials Pn(x) (Krall's polynomials) that are eigenfunctions of higher-order di erential operators. Using this method we show that recurrence coe cients of Krall's polynomials Pn(x) are rational functions of n. Let P (a; b; M ) n (x) be polynomials obtained from the Jacobi polynomials P (a; b) n (x) by the following procedure. We add an arbitrary concentrated mass M at the endpoint of the orthogonality interval with respect to the weight function of the ordinary Jacobi polynomials. We ÿnd necessary conditions for the parameters a; b in order for the polynomials P (a; b; M ) n (x) to obey a higher-order di erential equation. The main result of the paper is the following. Let a be a positive integer and b¿ -1=2 an arbitrary real parameter. Then the polynomials P (a; b; M ) n (x) are Krall's polynomials satisfying a di erential equation of order 2a + 4.