Lie-algebraic approach to the theory of polynomial solutions. I. Ordinary differential equations and finite-difference equations in one variable

Abstract

A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial element of the universal enveloping algebra of $s{l}_2(\mathbf{R})$ (for differential equations) or $s{l}_2(\mathbf{R}{)}_q$ (for finite-difference equations) in the "projectivized" representation possessing an invariant subspace. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

19pp

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