Related papers: Reciprocals of thinned exponential series

We show that the reciprocal of a partial sum with 2m terms of the alternating exponential series is the exponential generating function for permutations in which every increasing run has length congruent to 0 or 1 modulo 2m. More generally…

If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a+b, where a\in A and b\in B, in an even number of ways. We…

It is well known since A. J. Kempner's work that the series of the reciprocals of the positive integers whose the decimal representation does not contain any digit 9, is convergent. This result was extended by F. Irwin and others to deal…

Suppose l=2m+1, m>0. We introduce m "theta-series", [1],...,[m], in Z/2[[x]]. It has been conjectured that the n for which the coefficient of x^n in 1/[i] is 1 form a set of density 0. This is probably always false, but in certain cases,…

In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.

We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree $N$ whose Mahler measure is bounded by a constant. After a change of variables this reduces to a generalization of Ginibre's complex and real…

If \(A \) is a set of natural numbers containing \(0 \), then there is a unique nonempty "reciprocal" set \(B \) of natural numbers (containing \(0 \)) such that every positive integer can be written in the form \(a + b \), where \(a \in A…

An inverse polynomial has a Chebyshev series expansion 1/\sum(j=0..k)b_j*T_j(x)=\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the a_n are linear…

We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent…

The divisibility of truncated binomial series by their exponent n is analyzed. Divisibility is shown to depends on the divisibility characteristics of the integers constituting the binomials. Series division by the highest possible powers…

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product…

We show that sequences of positive integers whose ratios $a_n^2/a_{n+1}$ lie within a specific range are almost uniquely determined by their reciprocal sums. For instance, the Sylvester sequence is uniquely characterized as the only…

We consider the numbers $\mathcal{B}_{r,s} = (\mathbf{B}+1)^r \mathbf{B}^s$ (in umbral notation $\mathbf{B}^n = \mathbf{B}_n$ with the Bernoulli numbers) that have a well-known reciprocity relation, which is frequently found in the…

The concept of a fully interlacing matrix of formal power series with real coefficients is introduced. This concept extends and strengthens that of an interlacing sequence of real-rooted polynomials with nonnegative coefficients, in the…

In this article, we consider the reciprocal antisymmetric polynomial \[P(z) = \sum_{j = 0}^{s}(-1)^j\gamma_j\left(z^j - z^{N + s + 1 - j}\right), \ \gamma_0 = 1.\] It is shown that if all the zeros of $P(z)$ are located on the unit circle,…

The non-associative exponential series $exp(x)$ is a power series with monomials from the magma $M$ of finite, planar rooted trees. The coefficient $a(t)$ of $exp(x)$ relative to a tree $t$ of degree $n$ is a rational number and it is shown…

Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal…

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

Let $\alpha$ be an algebraic integer of degree $d$, which is reciprocal. The house of $\alpha$ is the largest modulus of its conjugates. We proved that $d$-th power of the house of reciprocal $\alpha$ has a limit point. We presented a…

Let $A=[a_{n,k}]_{n,k\ge 0}$ be an infinite lower triangular matrix defined by the recurrence $$a_{0,0}=1,\quad a_{n+1,k}=r_{k}a_{n,k-1}+s_{k}a_{n,k}+t_{k+1}a_{n,k+1},$$ where $a_{n,k}=0$ unless $n\ge k\ge 0$ and $r_k,s_k,t_k$ are all…