Related papers: An algorithm for evaluating the Gamma function and…
We explore a number of functional properties of the $q$-gamma function and a class of its quotients; including the $q$-beta function. We obtain formulas for all higher logarithmic derivatives of these quotients and give precise conditions…
This note introduces a new range of modified gamma and beta $k$ functions. The authors present new modified gamma and beta $k$-functions, first and second summation relations, various functionals, Mellin transforms, and integral…
In this paper we explore special values of Gaussian hypergeometric functions in terms of products of Euler $\Gamma$-functions and exponential functions of linear functions of the hypergeometric parameters. They include some classical…
An algorithm is given for computing explicit formulas for the generators of relations among the invariant rational functions for vector-valued bilinear forms. These formulas have applications in the geometry of Riemannian submanifolds and…
A $q$-analogue of the Hurwitz zeta-function is introduced through considerations on the spectral zeta-function of quantum group $SU_{q}(2)$, and its analytic aspects are studied via the Euler-MacLaurin summation formula. Asymptotic formulas…
We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power…
Taking the product of (2n+1)/(2n+2) raised to the power +1 or -1 according to the n-th term of the Thue-Morse sequence gives rise to an infinite product P while replacing (2n+1)/(2n+2) with (2n)/(2n+1) yields an infinite product Q, where P…
A new integral representation is derived using a definite integral given by Cauchy and used to evaluate a number of integrals containing the finite series of special functions.
We propose an algorithm for evaluation of the cumulative bivariate normal distribution, building upon Marsaglia's ideas for evaluation of the cumulative univariate normal distribution. The algorithm is mathematically transparent, delivers…
Using a self-replicating method, we generalize with a free parameter some Borwein algorithms for the number $\pi$. This generalization includes values of the Gamma function like $\Gamma(1/3)$, $\Gamma(1/4)$ and of course…
We introduce a novel approach to the automated termination analysis of computer programs: we use neural networks to represent ranking functions. Ranking functions map program states to values that are bounded from below and decrease as a…
In the paper, the authors review origins, motivations, and generalizations of a series of inequalities involving several exponential functions and sums, establish three new inequalities involving finite exponential functions and sums by…
The recurrence matrix relations, differentiation formulas, and analytical and fractional integral properties of incomplete gamma matrix functions $\gamma(Q, x)$ and $\Gamma(Q, x)$ are all covered in this article. The generalized incomplete…
We show how to calculate particular values of the Gamma function for specific rational arguments in the interval (0,1) without using the Elliptic K-function. Instead we use transcendental constants or periods defined by hyperelliptic…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
Given a gamma population with known shape parameter $\alpha$, we develop a general theory for estimating a function $g(\cdot)$ of the scale parameter $\beta$ with bounded variance. We begin by defining a sequential sampling procedure with…
We relate the singularities of a scheme $X$ to the asymptotics of the number of points of $X$ over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More…
This paper deals with evaluating constant terms of a special class of rational functions, the Elliott-rational functions. The constant term of such a function can be read off immediately from its partial fraction decomposition. We combine…
We present a uniform methodology for computing with finitely generated matrix groups over any infinite field. As one application, we completely solve the problem of deciding finiteness in this class of groups. We also present an algorithm…
Phase estimation is a quantum algorithm for measuring the eigenvalues of a Hamiltonian. We propose and rigorously analyse a randomized phase estimation algorithm with two distinctive features. First, our algorithm has complexity independent…