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Fourth-order Differential Equations Satisfied by the Generalized Co-recursive of all Classical Orthogonal Polynomials. A Study of Their Distribution of Zeros

A. Ronveaux
A. Zarzo
E. Godoy
Journal / Anthology

Journal of Computational and Applied Mathematics
Year: 1995
Volume: 59
Page range: 295-328

The unique fourth-order differential equation satisfied by the generalized co-recursive of all classical orthogonal polynomials is given for any (but fixed) level of recursivity. Up to now, these differential equations were known only for each classical family separately and also for a specific recursivity level. Moreover, we use this unique fourth-order differential equation in order to study the distribution of zeros of these polynomials via their Newton sum rules (i.e., the sums of powers of their zeros) which are closely related with the moments of such distribution. Both results are obtained with the help of two programs built in Mathematica symbolic language.

*Mathematics > Calculus and Analysis > Differential Equations
*Mathematics > Calculus and Analysis > Special Functions