Abstract
In this paper, we develop a general theory of quasi-orthogonal polynomials. We first derive three-term recurrence relation and second-order differential equations for quasi-orthogonal polynomials. We also give an expression for their discriminants in terms of the recursion coefficients of the corresponding orthogonal polynomials. In addition, we investigate an electrostatic equilibrium problem where the equilibrium position of movable charges is attained at the zeros of the quasi-orthogonal polynomials. The examples of the Freud weight w ( x ) = e − x 4 + 2 t x 2 and the Jacobi weight w ( x ) = ( 1 − x ) α ( 1 + x ) β are discussed in some detail. Finally, we consider the nonlinear orthogonality preserving transformation and related matrix problem.
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