# On the classification of differential equations having orthogonal polynomial solutions

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## 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)

_{m},(x)} that simultaneously satis]ies n distributionaldifferential equations of orders 1, 3, 5, ... (2

*n*−1). In particular, this weight ″ must satisfy

$$nb_{2n} (x)\Lambda ' + (nb_{2n}^\prime (x) - b_{2n - 1} (x))\Lambda = 0$$

_{ 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## Preview

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### References

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