Related papers: A divided difference identity for a class of multi…

This paper deals with properties of the algebraic variety defined as the set of zeros of a "typical" sequence of polynomials. We consider various types of "nice" varieties: set-theoretic and ideal-theoretic complete intersections,…

Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example,…

Differentiable real function reproducing primes up to a given number and having a differentiable inverse function is constructed. This inverse function is compared with the Riemann-Von Mangoldt exact expression for the number of primes not…

We introduce a class of doubly indexed real Hermite polynomials and we deal with their related properties like the associated recurrence formulae, Runge's addition formula, generating function and Nielsen's identity.

In 1954 it was proved if f is infinitely differentiable in the interval I and some derivative (of order depending on x) vanishes at each x, then f is a polynomial. Later it was generalized for multi-variable case. In this paper we give an…

We derive an identity involving Horadam numbers. Numerous new identities as well as those found in the existing literature are subsumed in this single identity.

We present an algorithm for computing a holonomic system for a definite integral of a holonomic function over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including…

We translate Uchimura's identity for the divisor function and whose generalizations into combinatorics of partitions, and give a combinatorial proof of them. As a by-product of their proofs, we obtain some combinatorial results.

We consider the known functional identity on the Weierstrass sigma function. A complete classification of odd entire functions which satisfy the same identity is obtained.

We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first example, to our knowledge, of a simply connected smooth variety whose sets of integral points are never…

An integro-differential ring is a differential ring that is closed under an integration operation satisfying the fundamental theorem of calculus. Via the Newton--Leibniz formula, a generalized evaluation is defined in terms of integration…

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided…

We establish an explicit inequality for the number of divisors of an integer $n$. It uses the size of $n$ and its number of distinct prime divisors.

The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of…

We use earlier defined notion of $n$- determinant to investigate sub-determinants of an extended Vandermonde matrix. Firstly, we demonstrate our method on a number of particular cases. Then we prove that all these results may be stated in…

We prove an important property of the binomial transform: it converts multiplication by the discrete variable into a certain difference operator. We also consider the case of dividing by the discrete variable. The properties presented here…

A new polynomial sieve is presented and used to show that almost all integers have at most one representation as a sum of two values of a given polynomial of degree at least 3.

Sums of the form $\sum_{q \leq N_1 < \cdots < N_m \leq n}{a_{(m);N_m}\cdots a_{(2);N_2}a_{(1);N_1}}$ date back to the sixteen century when Vi\`ete illustrated that the relation linking the roots and coefficients of a polynomial had this…

In this paper, we derive eight basic identities of symmetry in three variables related to Bernoulli polynomials and power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in…