ON NEWTON INTERPOLATING SERIES AND THEIR APPLICATIONS

Newton interpolating series are constructed by means of New- ton interpolating polynomials with coe±cients in an arbitrary field K (see Section 1). If K = C is the ¯field of complex numbers with the ordinary absolute value, particular convergent series of this form were used in number theory to prove the transcendence of some values of exponential series (see Theorem 1). Moreover, if K = Ҳ, by means of these series it can be obtained solutions of a multipoint boundary value problem for a linear ordinary differential equation (see Theorem 2). If K = Cp, some particular convergent series of this type (so-called Mahler series) are used to repre- sent all continuous functions from Zp in Cp (see [4]). For an arbitrary field K; with respect to suitable addition and multi- plication of two elements the set of Newton interpolating series becomes a commutative K-algebra KS[[X]] which generalizes the canonical K-algebra of formal power series. If we consider K a local. field, we construct a sub- algebra of KS[[X]], even for more variables, which is a generalization of Tate algebra used in rigid analytic geometry (see Section 3).