Related papers: Cancellation of factorials
The main aim of this paper is to derive some new summation theorems for terminating and truncated Clausen's hypergeometric series with unit argument, when one numerator parameter and one denominator parameter are negative integers. Further,…
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical…
We use the properties of Hermite and Kamp\'e de F\'eriet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher-order polynomial. The results we obtain are extended to product of…
Algebras of ultradifferentiable generalized functions are introduced. We give a microlocal analysis within these algebras related to the regularity type and the ultradifferentiable property.
Let $P$ be a prime polynomial in the variable $Y$ over a finite field and let $f$ be a quadratic irrational in the field of formal Laurant series in the variable $Y^{-1}$. We study the asymptotic properties of the degrees of the…
In this paper we establish properties of independence for the continued fraction expansions of two algebraic numbers. Roughly speaking, if the continued fraction expansions of two irrational algebraic numbers have the same long sub-word,…
We will see that key concepts of number theory can be defined for arbitrary operations. We give a generalized distributivity for hyperoperations (usual arithmetic operations and operations going beyond exponentiation) and a generalization…
In this note we present a refinement of the AM-GM inequality, and then we estimate in a special case the typical size of the improvement.
We present an application of elimination theory to the study of singularities over arbitrary fields, particularly to the open problem of resolution. A partial extension of a function, defining resolution of singularities over fields of…
We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental…
We develop number theoretic tools that allow to perform computations relevant for the quantum mechanics over finite fields of arbitrary, odd size, with the same speedup that is enjoyed by the Fast Fourier Transform.
The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of such functions with parameters whose values differ from the original ones by integers, is…
We show that degrees containing a complete extensions of arithmetic have the random join property: they are the supremum of any random real they compute, with another random real. The same is true for the truth-table and weak truth-table…
We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is…
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
We discuss algorithms for arithmetic properties of hypergeometric functions. Most notably, we are able to compute the p-adic valuation of a hypergeometric function on any disk of radius smaller than the p-adic radius of convergence. This we…
We obtain a lifting property for finite quotients of algebraic groups, and applications to the structure of these groups.
It is well-known that differentiation of hypergeometric function multiplied by a certain power function yields another hypergeometric function with a different set of parameters. Such differentiation identities for hypergeometric functions…
In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We…
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.