Related papers: A Balanced Three-term Generalization of Nicomachus…

Ramanujan stated an identity to the effect that if three sequences $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ are defined by $r_1(x)=:\sum_{n=0}^{\infty}a_nx^n$, $r_2(x)=:\sum_{n=0}^{\infty}b_nx^n$ and $r_3(x)=:\sum_{n=0}^{\infty}c_nx^n$ (here each…

We produce nontrivial asymptotic estimates for shifted sums of the form $\sum a(h)b(m)c(2m-h)$, in which $a(n),b(n),c(n)$ are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate…

We study the combinatorial properties of final types, which are certain non-decreasing sequences of integers, together with the partitions naturally associated with them. As a consequence, we obtain an identity expressing the $n$-nacci…

We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the…

We give a generalization of Collatz conjecture or 3n+1 problem on 2-adic completion of Q. A isometric of $Q_2$ provides information on the average behavior of the firsts terms of the sequence according to the class of $u_0$ modulo $2^m$. A…

We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these…

A one-to-one continuous function from a triangle to itself is defined that has both interesting number theoretic and analytic properties. This function is shown to be a natural generalization of the classical Minkowski ?(x) function. It is…

In this paper, we present the foundations of Summability Calculus, which places various established results in number theory, infinitesimal calculus, summability theory, asymptotic analysis, information theory, and the calculus of finite…

We consider a two-parameter family of triangles whose $(n,k)$-th entry (counting the initial entry as the $(0,0)$-th entry) is the number of tilings of $N$-boards (which are linear arrays of $N$ unit square cells for any nonnegative integer…

We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (\alpha n + \beta k + \gamma)\, T(n-1,k)+(\alpha' n + \beta' k + \gamma') \, T(n-1,k-1)$ with initial condition $T(0,k) = \delta_{k0}$ and parameters…

We extend the Reed Dawson identity for Knuth's old sum with a complex parameter, and we offer two separate hypergeometric series-based proofs of this generalization, and we apply this generalization to introduce binomial-harmonic sum…

In 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb{N}=\{0,1,\cdots\})$ with $x+3y$ a square. Meanwhile, he also…

An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown…

We examine what integers are representable as sums of three cubes. We also provide formulas for the number of representations of $x^3+y^3+z^3=n$ under the condition $x+y+z=t$. Also we show how the problem of three cubes is related to…

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…

In this paper, we introduce 3-dimensional $L-$summing method, which is a rearrangement of the summation $\sum A_{abc}$ with $1\leq a,b,c\leq n$. Applying this method on some special arrays, we obtain some identities on the Riemann zeta…

We provide a short proof of an algebraic identity. For integers $n\ge 2$ and variables $x,y,z$, it represents $(x^n+y^n-z^n)$ as a value of the quadratic form $\mathcal A^2+\mathcal B^2-\mathcal C^2$ after multiplication by an explicit…

This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an…

Simon's congruence, denoted \sim_n, relates words having the same subwords of length up to n. We show that, over a k-letter alphabet, the number of words modulo \sim_n is in 2^{\Theta(n^{k-1} log n)}.

For an arbitrary integer $x$, an integer of the form $T(x)=\frac{x^2+x}{2}$ is called a triangular number. For positive integers $\alpha_1,\alpha_2,\dots,\alpha_k$, a sum…