On a modification of the Jacobi linear functional: Asymptotic properties and zeros of the corresponding orthogonal polynomials
In this nicely written paper the authors treat the discrete Sobolev orthogonal polynomials generated by the linear functional \[ {\mathcal U}={\mathcal J}_{\alpha,\beta}+A_1\delta(x-1)+B_1\delta(x+1)-A_2\delta'(x-1)-B_2\delta'(x +1), \] where for a polynomial \(p\) the Jacobi functional is defined by \[ \langle{\mathcal J}_{\alpha,\beta},p\rangle=\int_{-1}^1 p(x)(1-x)^{\alpha}(1+x)^{\beta}dx,\;\alpha,\beta>-1. \] (Remark: the action of \(\mathcal U\) on a polynomial leads to \(+A_2p'(1)+B_2p'(-1)\) because of the derivative of the delta functional) After stating some results concerning the ordinary Jacobi polynomials (\(A_1=A_2=B_1=B_2=0\)) the authors give some properties of semi classical orthogonal polynomials `of class \(s\)' (a degree statement concerning the polynomial coefficients of the Pearson equation satisfied by the weight function). The sections with new results address problems on the values of the Jacobi kernels \[ K_n^{\alpha,\beta(p,q)}=\sum_{m=0}^n {(P_m^{(\alpha,\beta)})^{(p)}(x) (P_m^{(\alpha,\beta)})^{(q)}(y)\over \|P_m^{(\alpha,\beta)}\|^2}={\partial^{p+q}\over \partial x^p\partial y^q}K_n^{\alpha,\beta(0,0)}(x,y) \] at the points \((1,1)\) and \((1,-1)\), which play an important role in an existence theorem for orthogonal polynomials with respect to \(\mathcal U\). Moreover, problems on the order and class of the distributional differential equation for \(\mathcal U\), the hypergeometric character (a \({}_6F_5\)), asymptotic formulas (both in and outside the interval \((-1,1)\)) and the number of real zeros in this interval are studied. The paper concludes with a short discussion and some open problems.
scientific article; zbMATH DE number 1783467