Related papers: Subword complexity and Laurent series with coeffic…

A non-traditional proof of the Gregory-Leibniz series, based on the relationships among the zeta function, Bernoulli coefficients, and the Laurent expansion of the cotangent is given. New series for calculating pi are obtained.

The Laurent series expansions of zeta-functions play an important role in understanding their behavior near singularities, and their coefficients often encode significant arithmetic information. In the case of the Riemann and Hurwitz…

There is a theory of continued fractions for Laurent series in x^{-1} with coefficients in a field F. This theory bears a close analogy with classical continued fractions for real numbers with Laurent series playing the role of real numbers…

It is known, that among the formal solutions of the sixth Painlev\'e equation there met series with integer power exponents of the independent variable $x$ with coefficients in form of formal Laurent series (with finite main parts) in…

In this article, our aim is to extend the research conducted by Kurokawa and Wakayama in 2003, particularly focusing on the $q$-analogue of the Hurwitz zeta function. Our specific emphasis lies in exploring the coefficients in the Laurent…

The function $ \tan(\pi x / 2) / (\pi x / 2) $ is expanded into a Laurent series of $ 1 - x^2 $, where the coefficients are given explicitly as combinations of zeta function of even integers. This is used to achieve a sequence of upper and…

In this note, using an idea from \cite{Amo-Carrillo-Sanchez} we derive some new series representations involving $\zeta(2n)$ and Euler numbers. Using a well-known series representation for the Clausen function, we also provide some new…

We compute the Hilbert series of the graded algebra of regular functions on a symplectic quotient of a unitary circle representation. Additionally, we elaborate explicit formulas for the lowest coefficients of the Laurent expansion of such…

Let $K$ be the field of Laurent series with complex coefficients, let $\mathcal{R}$ be the inverse limit of the standard-graded polynomial rings $K[x_1, \ldots, x_n]$, and let $\mathcal{R}^{\flat}$ be the subring of $\mathcal{R}$ consisting…

Given $b=-A\pm i$ with $A$ being a positive integer, we can represent any complex number as a power series in $b$ with coefficients in $\mathcal A=\{0,1,\ldots, A^2\}$. We prove that, for any real $\tau\geq 2$ and any non-empty proper…

In this note, we develop transformation formulae and expansions for the log tangent integral, which are then used to derive series acceleration formulae for certain values of Dirichlet L-functions, such as Catalan's constant. The formulae…

A description of the properties of \L with complex characters is given. By using these, together with the more familiar \L with real characters, it is shown how certain two dimensional lattice sums, which previously could not be put into…

V.I. Arnold has recently defined the complexity of a sequence of $n$ zeros and ones with the help of the operator of finite differences. In this paper we describe the results obtained for almost most complicated sequences of elements of a…

We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…

Let $q>2$, and let $a$ and $b$ be two elements of the finite field $\mathbb{F}_q$ with $a\ne 0$. Carlitz represented the transposition $(0a)$ by a polynomial of degree $(q-2)^3$. In this note, we represent the transposition $(ab)$ by a…

We describe the formalisation in Coq of a proof that the numbers e and $\pi$ are transcendental. This proof lies at the interface of two domains of mathematics that are often considered separately: calculus (real and elementary complex…

We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials…

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. The Carlitz rank of a permutation polynomial is a important measure of complexity of the polynomial. In this paper we find the sharp lower bound for the weight of any permutation…

Let $\gg$ be the Lie algebra of a compact Lie group and let $\theta$ be any automorphism of $\gg$. Let $\gk$ denote the fixed point subalgebra $\gg^\theta$. In this paper we present LiE programs that, for any finite dimensional complex…

Revisiting the results by Winternitz [Symmetry in physics, CRM Proc. Lecture Notes 34, American Mathematical Society, Providence, RI, 2004, pp. 215-227], we thoroughly refine his classification of Lie subalgebras of the real order-three…