A symmetrized conjugacy scheme for orthogonal expansions

We establish a symmetrization procedure in a context of general orthogonal expansions associated with a second order differential operator $L$, a Laplacian'. Combined with a unified conjugacy scheme furnished in our earlier article it allows, via a suitable embedding, to associate a differential-difference Laplacian’ $\mathbb{L}$ with the initially given orthogonal system of eigenfunctions of $L$, so that the resulting extended conjugacy scheme has the natural classical shape. This means, in particular, that the related `partial derivatives’ decomposing $\mathbb{L}$ are skew-symmetric in an appropriate $L^2$ space and they commute with Riesz transforms and conjugate Poisson integrals. The results shed also some new light on the question of defining higher order Riesz transforms for general orthogonal expansions.