A method of constructing Krallʹs polynomials

We propose a method of constructing orthogonal polynomials Pn(x) (Krallʹs polynomials) that are eigenfunctions of higher-order differential operators. Using this method we show that recurrence coefficients of Krallʹs polynomials Pn(x) are rational functions of n. Let Pn(a,b;M)(x) be polynomials obtained from the Jacobi polynomials Pn(a,b)(x) by the following procedure. We add an arbitrary concentrated mass M at the endpoint of the orthogonality interval with respect to the weight function of the ordinary Jacobi polynomials. We find necessary conditions for the parameters a,b in order for the polynomials Pn(a,b;M)(x) to obey a higher-order differential equation. The main result of the paper is the following. Let a be a positive integer and b⩾−1/2 an arbitrary real parameter. Then the polynomials Pn(a,b;M)(x) are Krallʹs polynomials satisfying a differential equation of order 2a+4.