Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros

✍ Scribed by J. Arvesú; R. Álvarez-Nodarse; F. Marcellán; K. Pan

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 975 KB
• Volume: 90
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(98)00005-3

✦ Synopsis

We obtain an explicit expression for the Sobolev-type orthogonal polynomials {Q~} associated with the inner product /' {p,q) = p(x)q(x)p(x)dx+A,p(l)q(1)+B,p(-1)q(-1)+A2p'(1)q'(1)+B2p'(-l)q'(-l), I where p(x)= (I -x)~(1 + xf is the Jacobi weight function, e, ~> -1, A l, BI, A2, B2/>0 and p, q E P, the linear space of polynomials with real coefficients. The hypergeometric representation (6F5) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [-1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Qn(x). Such a zero is located outside the interval [-1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.