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Some properties of orthogonal polynomials satisfying fourth order differential equations. (English) Zbl 0820.33005

Starting from the fact that the Jacobi type, Laguerre type and Legendre type polynomials can be obtained as solutions of certain differential equation of the form

a 2 (x,n)f n '' (x)+a 1 (x,n)f n ' (x)+a 0 (x,n)f n (x)=0

where the coefficients depend not only on x, but also on n, and making use of a well-known equation concerning the zeros of the classical orthogonal polynomials [see the paper by the first author, SIAM J. Math. Anal. 18, 1664-1668 (1987; Zbl 0648.34021)], the authors derive some connections between the Jacobi, Laguerre and Legendre-type polynomials and the classical Jacobi, Laguerre and Legendre polynomials, respectively. Among other properties, certain bounds for the zeros of the Jacobi, Laguerre and Legendre-type polynomials are also obtained.

33C45Orthogonal polynomials and functions of hypergeometric type