Related papers: The formal series Witt transform

Formal power series come up in several areas such as formal language theory , algebraic and enumerative combinatorics, semigroup theory, number theory etc. This paper focuses on the set x R[[x]] consisting of formal power series with zero…

This note is concerned with series of the forms $\sum f(a^n)$ and $\sum f(n^{-a})$ where f(a) possesses a Mellin transform and $a > 1$ or $a<0$ respectively. Integral representations are derived and used to transform these series in several…

The non-trivial zeros of the Riemann zeta function and the prime numbers can be plotted by a modified von Mangoldt function. The series of non-trivial zeta zeros and prime numbers can be given explicitly by superposition of harmonic waves.…

For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a…

Matrices over the ring of formal power series are considered. Normal forms with respect to various sub-groups of the two-sided transformations are constructed. The construction is based on the special property of the action: it induces a…

For any two arithmetic functions $f,g$ let $\bullet$ be the commutative and associative arithmetic convolution $(f\bullet g)(k):=\sum_{m=0}^k \left( \begin{array}{c} k m \end{array} \right)f(m)g(k-m)$ and for any $n\in\mathbb{N},$…

In this contribution we generalize the classical Fourier Mellin transform [S. Dorrode and F. Ghorbel, Robust and efficient Fourier-Mellin transform approximations for gray-level image reconstruction and complete invariant description,…

We introduce a new class of simple Lie algebras $W(n,m)$ that generalize the Witt algebra by using "exponential" functions, and also a subalgebra $W^*(n,m)$ thereof; and we show each derivation of $W^*(1,0)$ can be written as a sum of an…

Let $\sigma_a^{(N)}(n)=\sum_{d^{N}|n}d^a$. An explicit transformation is obtained for the generalized Lambert series $\sum_{n=1}^{\infty}\sigma_{a}^{(N)}(n)e^{-ny}$ for Re$(a)>-1$ using the recently established Vorono\"i summation formula…

A generalised Weber function is given by $\w_N(z) = \eta(z/N)/\eta(z)$, where $\eta(z)$ is the Dedekind function and $N$ is any integer; the original function corresponds to $N=2$. We classify the cases where some power $\w_N^e$ evaluated…

The Witt algebra W_n is the Lie algebra of all derivations of the n-variable polynomial ring V_n=C[x_1, ..., x_n] (or of algebraic vector fields on A^n). A representation of W_n is polynomial if it arises as a subquotient of a sum of tensor…

We consider a generalized Mathieu series where the summands of the classical Mathieu series are multiplied by powers of a complex number. The Mellin transform of this series can be expressed by the polylogarithm or the Hurwitz zeta…

We extend the big and $p$-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the…

The convolved Fibonacci numbers F_j^(r) are defined by (1-z-z^2)^{-r}=\sum_{j>=0}F_{j+1}^(r)z^j. In this note some related numbers that can be expressed in terms of convolved Fibonacci numbers are considered. These numbers appear in the…

We study very smooth functions on the real line, namely Schwartz functions, that satisfy a finite identity relating their translates and a single modulation. Concretely, we assume there is a nontrivial linear combination of translates of…

Given a real closed polytope $P$, we first describe the Fourier transform of its indicator function by using iterations of Stokes' theorem. We then use the ensuing Fourier transform formulations, together with the Poisson summation formula,…

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility…

In this paper, we will extend the falling and rising factorial transforms \cite{ref. 1} which in this case every arbitrary function can be applied. Then, the properties of these transforms will be investigated and some corollaries will be…

Let $m, n$ be positive integers such that $m>1$ divides $n$. In this paper, we introduce a special class of piecewise-affine permutations of the finite set $[1, n]:=\{1, \ldots, n\}$ with the property that the reduction $\pmod m$ of $m$…

Let (Y_t: t > 0) be the STIT tessellation process. We show that for all polytopes W with nonempty interior and all a>1, the renormalized random sequence (a^n Y_{a^n}: n integer) induced in W, is a finitary factor of a Bernoulli shift. As a…