Hyperseries in the non-Archimedean ring of Colombeau generalized numbers

This article is the natural continuation of the paper: Mukhammadiev et al. Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring "Equation missing" of Robinson-Colombeau is non-Archimedean and Cauchy complete, a classical series $$\sum _{n=0}^{+\infty }a_{n}$$ ∑ n = 0 + ∞ a n of generalized numbers "Equation missing" is convergent if and only if $$a_{n}\rightarrow 0$$ a n → 0 in the sharp topology. Therefore, this property does not permit us to generalize several classical results, mainly in the study of analytic generalized functions (as well as, e.g., in the study of sigma-additivity in integration of generalized functions). Introducing the notion of hyperseries, we solve this problem recovering classical examples of analytic functions as well as several classical results.

Recursion formulas for certain quadruple hypergeometric functions

A remarkably large number of hypergeometric (and generalized) functions and a variety of their extensions have been presented and investigated in the literature by many authors. In this paper, we introduce five new hypergeometric functions in four variables and then establish several recursion formulas for these new functions. Some interesting particular cases and consequences of the main results are also considered.

On a basic mean value theorem with explicit exponents

We follow a paper by Sedunova regarding Vaughan’s basic mean value Theorem to improve and complete a more general demonstration for a suitable class of arithmetic functions as started by Cojocaru and Murty. As an application we derive a basic mean value theorem for the von Mangoldt generalized functions.