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

22 pages, 4 figures.-- MSC1991 codes: 33C45; 33A65; 42C05.-- Dedicated to Professor Mario Rosario Occorsio on his 65th birthday.MR#: MR1624329 (99h:33029)Zbl#: Zbl 0924.33006We obtain an explicit expression for the Sobolev-type orthogonal polynomials $\{Q_n\}$ associated with the inner product $\langle p,q\rangle=\int^1_{-1}p(x)q(x)\rho(x)dx+A_1p(1)q(1)+B_1p(-1)q(-1)+A_2p'(1)q'(1)+B_2p'(-1)q'(-1)$, where $\rho(x)=(1-x)^\alpha(1+x)^\beta$ is the Jacobi weight function, $\alpha,\beta>-1$, $A_1,B_1,A_2,B_2\geq 0$ and $p,q\in\bold P$, the linear space of polynomials with real coefficients. The hypergeometric representation $({}_6F_5)$ 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 $Q_n(x)$. Such a zero is located outside the interval $[-1,1]$. We deduce its dependence on the masses. Finally, the WKB analysis for the distribution of zeros is presented.The research of the first author (J.A.) was supported by a grant of Ministerio de Educación y Cultura (MEC) of Spain. The research of the three first authors (J.A., R.A.N. and F.M.) was supported by Dirección General de Enseñanza Superior (DGES) of Spain under Grant PB 96-0120-C03-01 and INTAS Project INTAS 93-0219 Ext.Publicad