A new denition of the incomplete beta function as a distribution- valued meromorphic is given and finite parts it its partial derivatives at singular values are calculated compared with formulas in literature.

In this paper we consider a simple linear recurrence sequence $ G_n defined over function field in one variable the of complex numbers. We prove an upper bound on indices n and m such that + G_m is S $-unit. This analogue already known results number fields.

In the first part we have shown that, for L2-approximation of functions from a separable Hilbert space in worst-case setting, linear algorithms based on function values are almost as powerful arbitrary if widths square-summable. That is, they achieve same polynomial rate convergence. this sequel, prove similar result Banach spaces and other classes functions.