On a modification of the Jacobi linear functional: Asymptotic properties and zeros of orthogonal polynomials

Abstract

The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional ${\scr U}$ $$ {\scr U}={\scr J}_{\alpha,\beta}+A_1\delta(x-1)+B_1\delta(x+1)- A_2\delta'(x-1)-B_2\delta'(x+1), $$ where ${\scr J}_{\alpha,\beta}$ is the Jacobi linear functional, i.e. $$ \langle{\scr J}_{\alpha,\beta},p\rangle=\int _{-1}p(x)(1-x) alpha (1+x) beta dx,\quad \alpha,\beta>-1,\ p\in {\Bbb P}, $$ and ${\Bbb P}$ is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in $(-1,1)$ (inner asymptotics) and ${\Bbb C}\sbs [-1,1]$ (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional ${\scr U}$ is a generalization of one studied by T. H. Koornwinder when $A_2=B_2=0$. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi-Markov function by a rational function with two double poles at $\pm 1$. The denominators of the $[n-1/n]$ Padé approximants are our orthogonal polynomials.