On orthogonal polynomials associated with rational perturbations of linear functional (Q2841516)

The authors study a linear functional \(u\) that satisfies the equation NEWLINE\[NEWLINE (x-a)(x-b) u = \lambda (x-c) v, NEWLINE\]NEWLINE where \(a,b,c\in \mathbb{C}\), \(\lambda\in \mathbb{C}\setminus\{0\}\), and \(v\) is a regular linear functional. After an introductory section on notation and preliminary results, an explicit necessary and sufficient condition for the regularity of \(u\) is given in Section 2. Moreover, the coefficients of the three-term recurrence relation satisfied by the new family of orthogonal polynomials is presented. Linear relations between the polynomials orthogonal to \(u\) and \(v\) are also analyzed. In Section 3, the stability of the semi-classical families is proved. In the final section, an illustrative example is given.