Differential and Difference Equations for Products of Classical Orthogonal Polynomials
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Factorization of differential equations has been intensively studied (see, for instance, , ). Less results are known for difference equations. In this publication we are not giving a general approach to the theory of factorization but rather present some observations and derive formulae for further use in reference books and for symbolic computations.
Several specific examples which arise from the theory of classsical orthogonal polynomials are studied. They have, to our mind, significance for practical applications in physics.
The paper is based to some extent on the ideas developed in other publications of the authors , ,  but the angle of view on the problem is different. In the first section differential equations are dealt with. In the second section, our studies are concentrated on difference equations. In both cases knowing the equation for orthogonal polynomials we derive equations for their products. These latter equations are of higher order than the starting ones, and polynomial solutions can be sought as solutions of multiparametric spectral problem .
KeywordsOrthogonal Polynomial Legendre Polynomial Shift Operator Hermite Polynomial Jacobi Polynomial
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