Related papers: Novel Approach to Infinite Products Using Multipli…
In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a na\"ive version of Andrews' anti-telescoping technique quite well. These new theorems also put to rest any notion…
A new basis for the polynomial ring of infinitely many variables is constructed which consists of products of Schur functions and Q-functions. The transition matrix from the natural Schur function basis is investigated.
This paper presents a reformulation of the Leibniz product rule as a finite sum that expresses the fractional derivative of the product of two differentiable functions. This paper then proves the cases for when the product consists of an…
A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…
In this paper we present a new identity and some of its variants which can be used for finding solutions while solving fractional infinite and finite series. We introduce another simple identity which is capable of generating solutions for…
We provide estimates for the convolution product of an arbitrary number of "resurgent functions", that is holomorphic germs at the origin of $C$ that admit analytic continuation outside a closed discrete subset of $C$ which is stable under…
We consider typical finite dimensional complex irreducible representations of a basic classical simple Lie superalgebra, and give a sufficient condition on when unique factorization of finite tensor products of such representations hold. We…
In this note, we extend to a composite modulo a recent result of Chan (2016) dealing with mean values of the product of an integer and its multiplicative inverse modulo a prime number.
Ultraproducts are a well-known tool in the classical model theory of first-order logic. We explore their uses in the context of finite model theory.
Let $R$ be a standard graded algebra over an infinite field $\mathbb{K}$ and $M$ a finitely generated $\ZZ$-graded $R$-module. Let $I_1,\ldots I_m$ be graded ideals of $R$. The functions $r(M/I_1^{a_1}\ldots I_m^{a_m}M)$ and…
Recently, Ono and the third author discovered that the reciprocals of the theta series $(q;q)_\infty^3$ and $(q^2;q^2)_\infty(q;q^2)_\infty^2$ have infinitely many closed formulas in terms of MacMahon's quasimodular forms $A_k(q)$ and…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
We state and prove product formulae for several generating functions for sequences $(a_n)_{n\ge0}$ that are defined by the property that $Pa_n+b^2$ is a square, where $P$ and $b$ are given integers. In particular, we prove corresponding…
Two kinds of infinite product representations for Vign\'eras multiple gamma function are presented. As an application of these formulas, a multiplication formula for the function is derived.
Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then,…
We describe a simple machinery which translates results on algebraic sums of sets of reals into the corresponding results on their cartesian product. Some consequences are: 1. The product of a meager/null-additive set and a strong measure…
We develop first steps in the study of factorizations of elements in ultraproducts of commutative cancellative monoids into irreducible elements. A complete characterization of the (multi-)sets of lengths in such objects is given. As…
In this paper, a class of smoothing modulus-based iterative method was presented for solving implicit complementarity problems. The main idea was to transform the implicit complementarity problem into an equivalent implicit fixed-point…
A novel approach is introduced for deriving exact solutions to nonlinear systems of ordinary differential equations. This method consists of four parts. In the initial part, the examined nonlinear differential equation system is transformed…
We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.