We give necessary and sufficient conditions for an orthogonal series to converge in the mean-squares a nonnegative function. present many examples applications, analysis probability. In particular, we Lancaster-type of expansion $% \sum_{n\geq 0}c_{n}\alpha _{n}(x)\beta _{n}(y)$ with two sets polynomials $\left\{ \alpha _{n}\right\} $ \beta means-squares bivariate study properties set $C(\alpha...

For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relation...