Annali di Matematica Pura ed Applicata

, Volume 138, Issue 1, pp 35–53 | Cite as

On the classification of differential equations having orthogonal polynomial solutions

  • Lance L. Littlejohn
Article

Summary

Suppose ϕ m (x) is a polynomial of degree in that satisfies the differential equation
$$\sum\limits_{i = 1}^{2n} {b_i } (x)y^{(i)} (x) = \lambda _m y(x), m = 0, 1, 2, ...$$
(1)
where n is some fixed integer ≧1. We show that, under certain conditions, there exists anorthogonalizing weight distribution for {ϕm,(x)} that simultaneously satis]ies n distributionaldifferential equations of orders 1, 3, 5, ... (2n−1). In particular, this weight ″ must satisfy
$$nb_{2n} (x)\Lambda ' + (nb_{2n}^\prime (x) - b_{2n - 1} (x))\Lambda = 0$$
in the distributional sense. As a corollary to this result, we get part of H. L. Krall's 1938classification theorem which gives necessary and sufficient conditions on the existenee of anOPS of solutions to (1) in terms of the moments and the coefficients of b i (x). To illustrate the theory, we consider all of the known OPS's to (1).In particular, new light is shed upon the problem of finding a real weight distribution for the Bessel polynomials.

Keywords

Differential Equation Weight Distribution 1938classification Theorem Polynomial Solution Distributional Sense 

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1978

Authors and Affiliations

  • Lance L. Littlejohn
    • 1
  1. 1.San Antonio

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