Orthonormal polynomials with generalized Freud-type weights

✍ Scribed by T. Kasuga; R. Sakai

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 344 KB
• Volume: 121
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1016/s0021-9045(02)00041-2

✦ Synopsis

We consider a certain generalized Freud-type weight W 2 rQ ðxÞ ¼ jxj 2r expðÀ2QðxÞÞ; where r4 À 1 2 and Q : R-R is even and continuous, Q 0 is continuous, Q 0 40 in ð0; NÞ; and Q 00 is continuous in ð0; NÞ: Furthermore, Q satisfies further conditions. Recently, Levin and Lubinsky have studied the sequence of orthonormal polynomials fP n ðW 2 Q ; xÞg N n¼0 with the Freud weight W 2 Q ðxÞ ¼ expðÀ2QðxÞÞ on R, and then they have obtained many interesting properties of P n ðW 2 Q ; xÞ [LL1]. We investigate the properties of P n ðW 2 rQ ; xÞ; which contain extensions of Levin and Lubinsky's results and improvements of Bauldry's results [Ba1,LL1].