Multivariate mixed orthogonal functions satisfying three term relations

We study a non trivial extension of orthogonal functions introduced in [1] to several variables. This kind of functions satisfy mixed orthogonality conditions in the sense that the inner product of functions of different parity order is computed by means of a moment functional, and the inner product of elements of the same parity order is computed by a modification of the original moment functional. Existence conditions, three term relations with matrix coefficients, a Favard-type theorem for this kind of functions are proved. A method for constructing bivariate hybrid orthogonal functions from univariate orthogonal polynomials and univariate orthogonal functions is presented. Finally, we give a complete description of a sequence of mixed orthogonal functions on the unit ball on $\mathbb{R}^2$, that includes, as particular case, the classical orthogonal ball polynomials.
This is a joint work with Cleonice F. Bracciali, from Universidade Estadual Paulista, Brazil.