  In this paper we provide a probabilistic representation of Lagrange's
identity which we use to obtain Papathanasiou-type variance expansions of
arbitrary order. Our expansions lead to generalized sequences of weights which
depend on an arbitrarily chosen sequence of (non-decreasing) test functions.
The expansions hold for arbitrary univariate target distribution under weak
assumptions, in particular they hold for continuous and discrete distributions
alike. The weights are studied under different sets of assumptions either on
the test functions or on the underlying distributions. Many concrete
illustrations for standard probability distributions are provided (including
Pearson, Ord, Laplace, Rayleigh, Cauchy, and Levy distributions).
