Eisenstein's theorem on power series expansions of algebraic functions

ALGEBRAIC INDEPENENCE OF CERTAIN FORMAL POWER SERIES (II)

We shall extend the results of [5] and prove that if f = Z o a x ? Z [[X]] is algebraic over Q (x), where a = 1, ƒ 1 and if ? , ? ,..., ? are p-adic integers, then 1 ? , ? ,..., ? are linkarly independent over Q if and only if (1+x) ,(1+x) ,…,(1+x) are algebraically independent over Q (x) if and only if f , f ,.., f are algebraically independent over Q (x)

ALGEBRAIC INDEPENDENCE OF CERTAIN FORMAL POWER SERIES (I)

We give a proof of the generalisation of Mendes-France and Van der Poorten's recent result over an arbitrary field of positive characteristic and then by extending a result of Carlitz, we shall introduce a class of algebraically independent series.

Basis Construction from Power Series Expansions of Value Functions

This paper explores links between basis construction methods in Markov decision processes and power series expansions of value functions. This perspective provides a useful framework to analyze properties of existing bases, as well as provides insight into constructing more effective bases. Krylov and Bellman error bases are based on the Neumann series expansion. These bases incur very large in...

Power series expansions for Mathieu functions with small arguments

Power series expansions for the even and odd angular Mathieu functions Sem(h, cos θ) and Som(h, cos θ), with small argument h, are derived for general integer values of m. The expansion coefficients that we evaluate are also useful for the calculation of the corresponding radial functions of any kind.

On the Complexity of Algebraic Power Series