Related papers: On the genesis of BBP formulas
For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…
The main purpose and motivation of this article is to create a linear transformation on the polynomial ring of rational numbers. A matrix representation of this linear transformation based on standard fundamentals will be given. For some…
In a base phi representation a natural number is written as a sum of powers of the golden mean $\varphi$. There are many ways to do this. How many? Even if the number of powers of $\varphi$ is finite, then any number has infinitely many…
Given an integer $b\geqslant 2$ and a set $P$ of prime numbers, the set $T_P $ of Toeplitz numbers comprises all elements of $[0,b[$ whose digits $(a_n)_{n\geqslant 1}$ in the base-$b$ expansion satisfy $a_n=a_{pn}$ for all $p\in P$ and…
In this paper we give a generalization of the main results in \cite{ab,ab1} about $b$-ary expansions of algebraic numbers. As a byproduct we get a large class of new transcendence criteria. One of our corollaries implies that $b$-ary…
In this article we introduce the concept of almost $\mathcal{P}$-numbers. We survey the existing results in literature for almost cyclic numbers and give characterizations for almost abelian and almost nilpotent numbers proving these two…
We introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed…
Let $b\ge 2$ be an integer. We show that the set of real numbers that are Poisson generic in base $b$ is $\boldsymbol{\Pi}^0_3$-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel…
The betweenness centrality (BC) is an important quantity for understanding the structure of complex large networks. However, its calculation is in general difficult and known in simple cases only. In particular, the BC has been exactly…
In this paper, we present a new method to derive formulas for the generating functions of interval orders, counted with respect to their size, magnitude, and number of minimal and maximal elements. Our method allows us not only to…
Permutations can be represented as linear combinations of natural numbers with different powers. In this paper, its coefficient matrix and inverse matrix is derived, and the results show the coefficient matrix is a lower triangular matrix…
This article is about Pi Formulas, infinite series of fractions which sum to multiples of Pi. Each such one can be associated with a unique set $S_k$ of rough numbers, where $k$ is a prime number. Given $S_k$ for any prime $k$, the set…
Using the machinery from the theory of Calabi-Yau differential equations, we find formulas for $1/\pi^2$ of hypergeometric and non-hypergeometric types.
We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…
We prove identities for six infinite series whose values involve linear combinations of $\pi$ and $\operatorname{ln} 2$, that do not appear in standard infinite series references.
In this note, we extend the connection between the hydrogen atom and $\pi$ to the number $e$ via the Lerch's transcendent.
By using definition of Golden derivative, corresponding Golden exponential function and Fibonomial coefficients, we introduce generating functions for Bernoulli-Fibonacci polynomials and related numbers. Properties of these polynomials and…
In the logic programming paradigm, it is difficult to develop an elegant solution for generating distinguishing formulae that witness the failure of open-bisimilarity between two pi-calculus processes; this was unexpected because the…
A modified Lagrange Polynomial is introduced for polynomial extrapolation, which can be used to estimate the equally spaced values of a polynomial function. As an example of its application, this article presents a prime-generating…