Set Of Formal Power Series Research Articles

The loop of formal power series with noncommutative coefficients under substitution

The set of formal power series with coefficients in an associative but noncommutative algebra becomes a loop with the substitution product. We study this loop by describing certain Lie and Sabinin algebras related to it. Some examples of Lie algebras satisfying the standard identities of degrees 5 and 6 appear naturally.

Coleman norm and trace operators for polynomial formal groups

In this paper are investigated a polynomial formal group analogues of the operators introduced by Coleman for the Lubin Tate formal group and the multiplicative formal group. Explicit constructions of the norm and trace operators for Laurent series are given, their main propirtes are checked. The eigenvalues and root values of these operators are also studied, and a homomorphism is constructed that connects the additive structure and the structure of the formal module on the set of formal power series.

Construction method for generating functions of special numbers and polynomials arising from analysis of new operators

The aim of this paper is to construct a new method related to a family of operators to define generating functions for special numbers and polynomials. With the help of this method, we investigate various properties of these special numbers and polynomials with their generating functions, functional equations, and differential equations. We also give some algorithms to calculate the values of these numbers and polynomials. Moreover, we introduce some fundamental properties of this operator. By applying integral method to this operator, we obtain integral formulas together with combinatorial sums. Moreover, by applying this operator, the Lagrange inversion formula and convolution formula to these generating functions, we derive some novel identities and relations including combinatorial sums and combinatorial numbers including these new numbers and polynomials, ie, the Apostol‐Bernoulli numbers, the Apostol‐Euler numbers, the Stirling numbers, the central factorial numbers, and the other special numbers and polynomials. In addition, we give some examples derived from relations between composita and this operator on the set of formal power series. Finally, we give complex form of the Fourier series for these generating functions. By using these Fourier series, we not only derive some series relations and trigonometric sums including trigonometric function and hyperbolic functions but also give an example for a differential equation.

SOME UMBRAL CHARACTERISTICS OF THE ACTUARIAL POLYNOMIALS

The utility of exponential generating functions is that they are relevant for combinatorial problems involving sets and sub- sets. Sequences of polynomials play a fundamental role in applied mathematics, such sequences can be described using the exponen- tial generating functions. The actuarial polynomials a ( ) n (x); n = 0; 1; 2; , which was suggested by Toscano, have the following ex- ponential generating function: 1 X n=0 a ( ) n (x) n! t n = exp(t +x(1 e t )): A linear functional on polynomial space can be identied with a formal power series. The set of formal power series is usually given the structure of an algebra under formal addition and multiplica- tion. This algebra structure, the additive part of which agree with the vector space structure on the space of linear functionals, which is transferred from the space of the linear functionals. The algebra so obtained is called the umbral algebra, and the umbral calculus is the study of this algebra. In this paper, we investigate some umbral representations in the actuarial polynomials.

On convergence sets of formal power series

The convergence set of a divergent formal power series f ( x 0 , ⋯ , x n ) is the set of all “directions” ξ ∈ P n along which f is absolutely convergent. We prove that every countable union of closed complete pluripolar sets in P n is the convergence set of some divergent series f. The (affine) convergence sets of formal power series with polynomial coefficients are also studied. The higher-dimensional analogs of the results of Sathaye (J Reine Angew Math 283:86–98, 1976 ), Lelong (Proc Am Math Soc 2:11–19, 1951 ), Levenberg and Molzon (Math Z 197:411–420, 1988 ), and of Ribon (Ann Scuola Norm Sup Pisa Cl Sci (5) 3:657–680 2004 ) are obtained.

A combinatorial equivalence relation for formal power series

In this paper we examine an equivalence relation on the set of formal power series with nonzero constant term. This is done both in terms of functional equations and also by interlacing two concepts from Riordan group theory, the A-sequence and the Bell subgroup. The best known example gives an equivalence class{1+z,1/(1−z),C(z),T(z),Q(z),…} where C(z),T(z) and Q(z) are generating functions of the Catalan numbers, ternary numbers and quaternary numbers, respectively. A power series for one member of an equivalence class can be transformed into power series for the rest of members in the equivalence class and interpretations in terms of weighted lattice paths can also be given.

Fine costs for Euclid's algorithm on polynomials and Farey maps

This paper studies digit-cost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field Fq. The usual bit-complexity is defined with respect to the degree of the quotients; we focus here on a notion of ‘fine’ complexity (and on associated costs) which relies on the number of their non-zero coefficients. It also considers and compares the ergodic behavior of the corresponding costs for truncated trajectories under the action of the Gauss map acting on the set of formal power series with coefficients in a finite field. The present paper is thus mainly interested in the study of the probabilistic behavior of the corresponding random variables: average estimates (expectation and variance) are obtained in a purely combinatorial way thanks to classical methods in combinatorial analysis (more precisely, bivariate generating functions); some of our costs are even proved to satisfy an asymptotic Gaussian law.We also relate this study with a Farey algorithm which is a refinement of the continued fraction algorithm for the set of formal power series with coefficients in a finite field: this algorithm discovers ‘step by step’ each non-zero monomial of the quotient, so its number of steps is closely related to the number of non-zero coefficients. In particular, this map is shown to admit a finite invariant measure in contrast with the real case. This version of the Farey map also produces mediant convergents in the continued fraction expansion of formal power series with coefficients in a finite field.

ON THE ANALYTIC PROPERTIES OF THE z-COLOURED JONES POLYNOMIAL

We analyse the possibility of defining ℂ-valued Knot invariants associated with infinite-dimensional unitary representations of SL(2,ℝ) and the Lorentz Group taking as starting point the Kontsevich integral and the notion of infinitesimal character. This yields a family of knot invariants whose target space is the set of formal power series in ℂ, which contained in the Melvin–Morton expansion of the coloured Jones polynomial. We verify that for some knots the series have zero radius of convergence and analyse the construction of functions of which this series are asymptotic expansions by means of Borel re-summation. Explicit calculations are done in the case of torus knots which realise an analytic extension of the values of the coloured Jones polynomial to complex spins. We present a partial answer in the general case.

Stability and buchberger criterion for standard bases in power series rings

We prove that the property of a finite set of formal power series to be a standard basis of the ideal it generates is locally stable in the space of admissible term orders. Consequently, universal standard bases exist. We give a criterion for an ideal basis to be a standard basis which is an analogue to Buchberger's ‘critical pair criterion’ for Groebner bases.