On polynomial solutions of differential equations

Turbiner, A.

doi:10.1063/1.529848

On polynomial solutions of differential equations

A. Turbiner
(
- Marseille, CPT and
- Provence U.
)

Nov, 1991

8 pages

Published in:

J.Math.Phys. 33 (1992) 3989-3993

e-Print:

hep-th/9209077 [hep-th]

DOI:

10.1063/1.529848

Report number:

CPT-91-P-2628

View in:

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Abstract:

A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the "projectivized" representation possessing an invariant subspace and the spectral problem for a certain linear differential operator with variable coefficients. It is shown in general that polynomial solutions of partial differential equations occur; in the case of Lie superalgebras there are polynomial solutions of some matrix differential equations, quantum algebras give rise to polynomial solutions of finite--difference equations. Particularly, known classical orthogonal polynomials will appear when considering

SL(2,{\bf R})

acting on

{\bf RP_1}

. As examples, some polynomials connected to projectivized representations of

sl_2 ({\bf R})

,

sl_2 ({\bf R})_q

,

osp(2,2)

and

so_3

are briefly discussed.

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Figures(0)

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