Related papers: Certain results on Euler-type integrals and their …
Several integrals involving powers and ordinary hypergeometric functions are rederived by means of a generalized hypergeometric function of two variables (Appell's function) recovering some well-known expressions as particular cases. Simple…
Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in…
This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.
In this article, we express solutions of the Gauss hypergeometric equation as a series of the multiple polylogarithms by using iterated integral. This representation is the most simple case of a semisimple representation of solutions of the…
The definition for the $p$-adic Hurwitz-type Euler zeta functions has been given by using the fermionic $p$-adic integral on $\mathbb Z_p$. By computing the values of this kind of $p$-adic zeta function at negative integers, we show that it…
The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences $(1/2,1/4,1/4)$, $(1/2,1/3,1/6)$ and $(1/3,1/3,1/3)$. These form a special class of algebraic transformations of Gauss…
Expressions for a family of integrals involving the Hurwitz zeta function are established using standard properties of the Fourier transform.
We review Euler's idea on the Gammafunction. We will explain, how Euler obtained them and how Euler's ideas anticipate more modern approaches and theories. Furthermore, some questions asked by Euler are answered.
The functional relation of the Hurwitz zeta function is proved by using the connection problem of the confluent hypergeometric equation.
This paper discusses the incomplete Gamma and Beta integrals involving the generalised hypergeometric function. The distribution of the largest and the smallest roots of a ratio arising in comparing the mean differences among groups is…
We show how the combined use of the generating function method and of the theory of multivariable Hermite polynomials is naturally suited to evaluate integrals of gaussian functions and of multiple products of Hermite polynomials.
This article surveys the Euler calculus - an integral calculus based on Euler characteristic - and its applications to data, sensing, networks, and imaging.
We obtain new inequalities for certain hypergeometric functions. Using these inequalities, we deduce estimates for the hyperbolic metric and the induced distance function on a certain canonical hyperbolic plane domain.
The result of performing integrations over connection type variables in the path integral for the discrete field theory may be poorly defined in the case of non-compact gauge group with the Haar measure exponentially growing in some…
We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring…
The theories of hypergeometric functions and modular forms are highly intertwined. For example, particular values of truncated hypergeometric functions and hypergeometric character sums are often congruent or equal to Fourier coefficients…
We give a combinatorial interpretation for the hypergeometric functions associated with tuples of rational numbers.
We study the linear Pfaffian systems satisfied by a certain class of hypergeometric functions, which includes Gau\ss's ${}_2 F_{1}$, Thomae's ${}_L F_{L-1}$ and Appell-Lauricella's $F_D$. In particular, we present a fundamental system of…
In this paper,using methods of weight functions and techniques of real analysis, we provide a multidimensional Hilbert-type integral inequality with a homogeneous kernel of degree 0 as well as a best possible constant factor related to the…
The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: {\sigma}(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of…