Related papers: A congruence sum and rational approximations
We associate to a full flag $\mathcal{F}$ in an $n$-dimensional variety $X$ over a field $k$, a "symbol map" $\mu_{\mathcal{F}}:K(F_X) \to \Sigma^n K(k)$. Here, $F_X$ is the field of rational functions on $X$, and $K(\cdot)$ is the…
Two numbers $m$ and $n$ are considered amicable if the sum of their proper divisors, $s(n)$ and $s(m)$, satisfy $s(n) = m$ and $s(m) = n$. In 1981, Pomerance showed that the sum of the reciprocals of all such numbers, $P$, is a constant. We…
We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which…
We study approximation in the unit interval by rational numbers whose numerators are selected randomly with certain probabilities. Previous work showed that an analogue of Khintchine's Theorem holds in a similar random model and raised the…
We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using…
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…
In this paper we introduce a family of rational approximations of the reciprocal of a $\phi$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The…
We primarily investigate congruences modulo $p$ for finite sums of the form $\sum_k\binom{rk}{k}x^k/k$ over the ranges $0<k<p$ and $0<k<p/r$, where $p$ is a prime larger than the positive integer $r$. Here $x$ is an indeterminate, thus…
We derive and prove an explicit formula for the sum of the fractional parts of certain geometric series. Although the proof is straightforward, we have been unable to locate any reference to this result. This summation formula allows us to…
We give some results and conjectures about recurrence relations for certain sequences of binomial sums.
We consider general subordination and obtain the formula of the subordinated predictable compensator. An example of application is given.
It was shown in that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from…
We introduce an explicit formula for a reciprocal sum related to the Riemann zeta function at s=6, and pose one question related to a computational formula for larger values of s.
We give a short, self-contained proof of Stanley's reciprocity theorem for a rational cone K \subset R^d. Namely, let sigma_K (x) = sum_{m \in K \cap Z^d} x^m. Then sigma_K (x) and sigma_int(K) (x) are rational functions which satisfy the…
This paper describes a mathematical formulation for measuring how one system can estimate the consciousness of another. This consciousness estimate is always relative to the observer. The paper shows how this formulation leads to simple…
A finite sum of exponential functions may be expressed by a linear combination of powers of the independent variable and by successive integrals of the sum. This is proved for the general case and the connection between the parameters in…
In this paper, we investigate some congruences involving sums of $\frac{d^{-k}{x\choose k}{x+k\choose k}}{{2k \choose k}}$, where $x$ be a $p$-adic integer, $k$ be a non-negative integer, and $d$ $(d\neq 0)$ be a rational number.
The theory of quaternionic modular forms has been studied for decades as an example of the modular forms of many variables. The purpose of this study is to provide some congruence relations satisfied by such quaternionic modular forms.
We give a bijective parameter representation for a sum of squares of numbers being equal to another sum of squares of numbers.
Computing the probability of a formula given the probabilities or weights associated with other formulas is a natural extension of logical inference to the probabilistic setting. Surprisingly, this problem has received little attention in…