A product representation for power series

We investigate the translation equation (T) , and the cocycle equations (Co1) , (Co2) in [ x ], the ring of formal power series over . Here we restrict ourselves to iteration groups of type I, i. e. to solutions of (T) of the form , where c 1 ? 1 is a generalized exponential function. It is easy to prove that the coefficient functions c n (s), a n (s), s n (s) of are polynomials, c 1(s), polynomials or rational functions in one more generalized exponential function and, in a special case, polynomials in an additive function. For example, we obtain . For this we do not need any detailed information about the polynomials P n . Under some conditions on the exponential and additive functions it is possible to replace the exponential and additive functions by independent indeterminates. In this way we obtain formal versions of the translation equation and the cocycle equations in rings of the form [ x ], [ x ], [ x ], [ x ], and [ x ]. We solve these equations in a completely algebraic way, by deriving formal differential equations or Aczel�Jabotinsky type equations for these problems. It is possible to get the structure of the coefficients in great detail which are now polynomials or rational functions. We prove the universal character (depending on certain parameters) of these polynomials or rational functions. And we deduce the canonical form S -1(yS(x)) for iteration groups of type I. This approach seems to be more clear and more general than the original one. Some simple substitutions allow us to solve these problems in rings of the form ( [ u ]) [ x ], i. e. where the coefficient functions are formal series.

Formal power series and regular operations on fuzzy languages

In this article, we have established a relation between total number of partitions of a positive integer n and all possible factorizations of a power series with constant term prime power pn, into irreducible power series. Finally we try to develop an irreducibility criterion for power series whose constant term is a prime power.

The constant term of certain rational functions has attracted much attention recently. For example the Dyson conjecture; that the constant term ofis the multinomial coefficienthas spawned many generalizations (see [2], [7]). In this paper we consider some other families of rational functions which have interesting constant terms. For example, Corollary 4 states that the constant term of(1.1)is . Here, and throughout this paper, A and B denote fixed positive integers.In order to prove this result, we consider the rational function in two variables

Formal analysis is the study of formal power series, formal Laurent series, formal root series, and other formal series or formal functionals. This book is the first comprehensive presentation of the topic that systematically introduces formal analysis, including its algebraic, analytic, and topological structure, along with various applications.

Using methods analogous to those in diophantine approximation, the existence of good rational (function) approximations to the general solutions of each of a class of linear homogeneous differential equations is investigated. It is shown that, while many good approximations may exist, no truly extraordinary approximations exist. The proofs are via Lemmas on formal series which remain valid in a much wider context, e. g. even if the field of constants has positive characteristic.

Reduction of everywhere convergent power series with respect to Gröbner bases

In this chapter, we consider a system of differential equations $$x\frac{{d\vec{y}}}{{dx}} = \vec{f}\left( {x,\vec{y}} \right)$$ (E) assuming that the entries of the \({{\mathbb{C}}^{n}}\)-valued function \(\vec{f}\) are convergent power series in complex variables \(\left( {x,\vec{y}} \right) \in {{\mathbb{C}}^{{n + 1}}}\) with coefficients in ℂ, where x is a complex independent variable and \(\vec{y} \in {{\mathbb{C}}^{n}}\) is an unknown quantity. The main tool is calculation with power series in x. In §I-4, using successive approximations, we constructed power series solutions. However, generally speaking, in order to construct a power series solution \(\vec{y}\left( x \right) = \sum\limits_{{m = 1}}^{\infty } {{{x}^{m}}{{{\vec{a}}}_{m}}}\) this expression is inserted into the given differential equation to find relationships among the coefficients \({{\vec{a}}_{m}}\), and the coefficients \({{\vec{a}}_{m}}\) are calculated by using these relationships. In this stage of the calculation, we do not pay any attention to the convergence of the series. This process leads us to the concept of formal power series solutions (cf. §V-1). Having found a formal power series solution, we estimate \(|{{\vec{a}}_{m}}|\) to test its convergence. As the function \({{x}^{{ - 1}}}\vec{f}\left( {x,\vec{y}} \right)\) is not analytic at x = 0, Theorem I-4-1 does not apply to system (E). Furthermore, the existence of formal power series solutions of (E) is not always guaranteed. Nevertheless, it is known that if a formal power series solution of (E) exists, then the series is always convergent. This basic result is explained in §V-2 (cf. [CL, Theorem 3.1, pp. 117-119] and [Wasl, Theorem 5.3, pp. 22-25] for the case of linear differential equations]). In §V-3, we define the S-N decomposition for a lower block-triangular matrix of infinite order. Using such a matrix, we can represent a linear differential operator $$ \mathcal{L}\left[ {\vec y} \right]{\text{ = }}x\frac{{d\vec y}}{{dx}}{\text{ + }}\Omega {\text{(}}x{\text{)}} \vec y, $$ (LDO) Where \(\Omega \left( x \right)\) is an n x n matrix whose entries are formal power series in x with coefficients in ℂ n . In this way, we derive the S-N decomposition of L in §V-4 and a normal form of L in §V-5 (cf. [HKS]). The S-N decomposition of L was originally defined in [GerL]. In §V-6, we calculate the normal form of a given operator L by using a method due to M. Hukuhara (cf. [Si17, §3.9, pp. 85-89]). We explain the classification of singularities of homogeneous linear differential equations in §V-7. Some basic results concerning linear differential equations given in this chapter are also found in [CL, Chapter 4].

Chapter 2 - Series

We use an elementary method to draw analytic conclusions from divergent formal power series solutions of systems of differential equations containing a parameter and give some applications to the theory of turning points. Our main result shows that a divergent formal series transformation of one system into another in which the coefficients satisfy certain estimates is necessarily the asymptotic expansion of an actual transformation. We use it to show the following. Given a two dimensional system ePdy/dx = A(x,e)y with A holomorphic at (x0,0), suppose that x0 is formally not a turning point in the sense that no singularities appear at x0 during the standard formal solution procedure with formal fractional power series in e. Then the formal solution is necessarily a uniform asymptotic representation of a fundamental matrix of the system on a full neighborhood of x0. (This conclusion is known to fail under weaker hypotheses on A). We also obtain similar but less complete results for higher order systems under more specialized hypotheses.

Substituting non-unit formal power series into formal power series is a well-understood concept. This article describes a sense in which one can define substitution of unit formal power series, implemented in Maple’s MultivariatePowerSeries package.

Let S n be the symmetric group, T d be the subspace of C[x 1 , x 2 ,…,x n ] which contains all of the homogeneous polynomials of degree d. The representation of S n determined by T d is denoted by λ d , and the irreducible representation characterized by partition a is denoted by π(a). We denote the multiplicity of π(a) entries into λ d by N a d . As an extension of [4], this note simplifies the recursion formula of the formal power series (?)N a d t d , and obtains the explicity expressions of generating function...

By multiplying a truncated aberration series by an appropriate, low-order polynomial containing a given number of terms, it is possible to produce a new power series for which a significantly larger number of the high-order coefficients are markedly smaller than those of the original power series. This indicates that, to a reasonable approximation, the higher-order coefficients form simple patterns that can be expected to extend into the unknown coefficients. This suggests the possibility of estimating unknown coefficients in order to reduce the error incurred by simply ignoring them. This is effectively realized when the original power series is replaced with the quotient of the new power series (which is expected to have a smaller truncation error) and the polynomial, yielding what is, for most purposes, a rational function. In this way it is possible to gain significantly greater advantage from the higher-order coefficients (say, beyond the first few orders) than was previously obtained. These rational functions are generalizations of the well-known Pade approximants for power series in one variable. Some numerical examples are presented that demonstrate the remarkable effectiveness of this method.

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series.

This paper deals with two-scale difference equations having a formal power series as symbol. We require that the equation has non-zero distributional solutions which are either compactly supported or integrals of compactly supported distributions with support bounded to the left. Such solutions are called eigenfunctions. As main result we determine the necessary and sufficient condition for the existence of eigenfunctions that the symbol must be a rational function with a special structure. We show that the eigenfunctions can be expressed by means of a finite sum of shifted eigenfunctions belonging to the case with a polynomial symbol (characteristic polynomial), which is well investigated.