The paper is devoted to study the $H$-function defined by the Mellin-Barnes integral
$$H^{m,n}_{\thinspace p,q}(z)={\frac1{2πi}}\int_{\Lss} \HHs^{m,n}_{\thinspace p,q}(s)z^{-s}ds,$$
where the function $\HH^{m,n}_{\thinspace p,q}(s)$ is a certain ratio of products of Gamma functions with the argument $s$ and the contour $\LL$ is specially chosen. The conditions for the existence of $H^{m,n}_{\thinspace p,q}(z)$ are discussed and explicit power and power-logarithmic series expansions of $H^{m,n}_{p,q}(z)$ near zero and infinity are given. The obtained results define more precisely the known results.
Kummer's Fourier series for the log gamma function is well known, having been discovered in 1847. In this paper we develop a corresponding Fourier series for the logarithm of the Barnes double gamma function (and the method may be easily extended to higher order multiple gamma functions). Some applications of these Fourier series are explored.
New proofs of the duplication formulae for the gamma and the Barnes double gamma functions are derived using the Hurwitz zeta function. Concise derivations of Gauss's multiplication theorem for the gamma function and a corresponding one for the double gamma function are also reported. This paper also refers to some connections with the Stieltjes constants.
In this paper we determine the Fourier series expansion of the log-Barnes function. This is the analogue of the classical result of Kummer and Malmsten. Applying this expansion we get some integrals similar to the Espinosa-Moll log-Gamma integrals with respect to $\log G$. During the course of the paper some interesting log-Gamma integrals and a log-harmonic sum are evaluated.
We examine a family ${}_pG_{q}^{\mathbb C}\big[\genfrac{}{}{0pt}{}{(a)}{(b)};z\big]$ of integrals of Mellin-Barnes type over the space ${\mathbb Z}\times {\mathbb R}$, such functions $G$ naturally arise in representation theory of the Lorentz group. We express ${}_pG_{q}^{\mathbb C}(z)$ as quadratic expressions in the generalized hypergeometric functions ${}_{p}F_{q-1}$ and discuss further properties of the functions ${}_pG_{q}^{\mathbb C}(z)$.
In this paper we establish new integral representations for the remainder term of the known asymptotic expansion of the logarithm of the Barnes $G$-function. Using these representations, we obtain explicit and numerically computable error bounds for the asymptotic series, which are much simpler than the ones obtained earlier by other authors. We find that along the imaginary axis, suddenly infinitely many exponentially small terms appear in the asymptotic expansion of the Barnes $G$-function. Employing one of our representations for the remainder term, we derive an exponentially improved asymp…
In many cases one may encounter an integral which is of $q$-Mellin--Barnes type. These integrals are easily evaluated using theorems which have a long history dating back to Slater, Askey, Gasper, Rahman and others. We derive some interesting $q$-Mellin--Barnes integrals and using them we derive transformation and summation formulas for nonterminating basic hypergeometric functions. The cases which we treat include ratios of theta functions, the Askey--Wilson moments, nonterminating well-poised ${}_3ϕ_2$, nonterminating very-well-poised ${}_5W_4$, ${}_8W_7$, products of two nonterminating ${}_…
We consider Mellin-Barnes integral representations of GKZ hypergeometric equations. We construct integration contours in an explicit way and show that suitable analytic continuations give rise to a basis of solutions.
We introduce completely monotonic functions of order $r>0$ and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order.
In this paper we provide a unified approach to a family of integrals of Mellin--Barnes type using distribution theory and Fourier transforms. Interesting features arise in many of the cases which call for the application of pull-backs of distributions via smooth submersive maps defined by Hörmander. We derive by this method the integrals of Hecke and Sonine relating to various types of Bessel functions which have found applications in analytic and algebraic number theory.
This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for high-precision computation of the Barnes gamma function and Glaisher's constant are also discussed.
We generalize our previous new definition of Euler Gamma function to higher Gamma functions. With this unified approach, we characterize Barnes higher Gamma functions, Mellin Gamma functions, Barnes multiple Gamma functions, Jackson $q$-Gamma function, and Nishizawa higher $q$-Gamma functions. This approach extends to more general functional equations. This generalization reveals the multiplicative group structure of solutions of the functional equation that appears as a cocycle equation. We also generalize Barnes hierarchy of higher Gamma function and multiple Gamma functions. In this new app…
We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta function. Especially, we prove that the bilateral zeta function has a nice Fourier series expansion and the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. By these properties of the bilateral zeta functions, We obtain simple proofs of some formulas, for example the reflection formula for the multiple gamma function, the inversio…
We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes Gamma function, including the particular case also known as multiple Gamma function.
Summations and relations involving the Hurwitz and Riemann zeta-functions are extended first to Barnes zeta-functions and then to zeta-functions of general type. The analysis is motivated by the evaluation of determinants on spheres which are treated both by a direct expansion method and by regularised sums. Comments on existing calculations are made. A Kaluza--Klein technique is introduced providing a determinant interpretation of the Glaisher-Kinkelin-Bendersky constants which are then generalised to arbitrary zeta-functions. This technique allows an improved treatment of sphere determinants…
The classical Lerch's formula states the following normalized product: $$\prod_{n=0}^\infty(x+n)=\frac{\sqrt{2π}}{Γ(x)},\quad \textrm{Re}(x)>0, $$ where $Γ(x)$ is the Euler gamma function. In this note, by using Barnes' multiple zeta function and its alternating form, we obtain two kinds of generalizations of Lerch's formula, which imply the product $$ \prod_{n=1}^\infty n=\sqrt{2π} $$ (in the sense of zeta regularization) and the product $$\frac{2\cdot2}{1\cdot 3}\frac{4\cdot4}{3\cdot 5}\frac{6\cdot6}{5\cdot 7}\cdots=\fracπ{2}$$ (Wallis' formula in 1656), respectively.
The bilinear maximal operator defined below maps $L^p\times L^q$ into $L^r$ provided $1<p,q<\zI$, $1/p+1/q=1/r$ and $2/3<r\le1$. $$ Mfg(x)=\sup_{t>0}\frac1{2t}\int_{-t}^t\abs{f(x+y)g(x-y)} dy.$$ In particular $Mfg$ is integrable\thinspace if $f$ and $g$ are square integrable, answering a conjecture posed by Alberto Calderón.
We establish a braid monodromy representation of functions satisfying an algebraic equation containing three terms (trinomial equation). We follow global analytic continuation of the roots to a trinomial algebraic equation that are expressed by Mellin-Barnes integral representations. We depict braids that arise from the monodromy around all branching points. The global braid monodromy is described in terms of rational twists of strands that yield a classical Artin braid representation of algebraic functions. As a corollary, we get a precise description of the Galois group of a trinomial algebr…
Integral transforms $$(\mbox{\boldmath$H$}f)(x)=\int^\infty_0H^{m,n}_{\thinspace p,q} \left[xt\left|\begin{array}{c}(a_i,α_i)_{1,p}\\[1mm](b_j,β_j)_{1,q} \end{array}\right.\right]f(t)dt$$ involving Fox's $H$-functions as kernels are studied in the spaces $\Ls_{ν,r}$ of functions $f$ such that $$\int^\infty_0|t^νf(t)|^r\frac{dt}t<\infty\quad(1\ \eqls\ r<\infty, \ ν\in\Rs).$$ Mapping properties such as the boundedness, the representation and the range of the transforms \boldmath$H$ are given.
The paper is devoted to study the inversion of the integral transform $$(\mbox{\boldmath$H$}f)(x)=\int^\infty_0H^{m,n}_{\thinspace p,q} \left[xt\left|\begin{array}{c}(a_i,α_i)_{1,p}\\[1mm](b_j,β_j)_{1,q} \end{array}\right.\right]f(t)dt$$ involving the $H$-function as the kernel in the space $\euf114_{ν,r}$ of functions $f$ such that $$\int^\infty_0\left|t^νf(t)\right|^r\frac{dt}t<\infty\quad(1<r<\infty, \ ν\in\msb122).$$
Let $G_n$ be the Barnes multiple Gamma function of order $n$ and the function $f_n(z)$ be defined as \begin{align*} f_n(z)=\dfrac{\log G_n(z+1)}{z^n\Log z},\quad z\in \mathbb{C}\setminus (-\infty,0]. \end{align*} In this work, a conjecture to find the Stieltjes representation is proposed such that $f_n(z)$ is a Pick function. The conjecture is established for the particular case $n=3$ by examining the properties of $f_3(z)$.
Our aim in this paper, is to establish certain new integrals for the the (p,q)-Mathieu--power series. In particular, we investigate the Mellin-Barnes type integral representations for a particular case of thus special function. Moreover, we introduce the notion of the (p,q)-Mittag-Leffler functions and we present a relationships between thus two functions. Some other applications are proved, in particular two Turán type inequalities for the (p,q)-Mathieu series are proved.
Euler's Gamma function $Γ$ either increases or decreases on intervals between two consequtive critical points. The inverse of $Γ$ on intervals of increase is shown to have an extension to a Pick-function and similar results are given on the intervals of decrease, thereby answering a question by Uchiyama. The corresponding integral representations are described. Similar results are obtained for a class of entire functions of genus 2, and in particular integral representations for the double gamma function and the $G$-function of Barnes are found.
The multiple gamma function $Γ_n$, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the $Γ_n$ function and their applications to summation of series and infinite products.
We consider the q-hypergeometric equation with q^{N}=1 and $α, β, γ\in {\Bbb Z}$. We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the subspace of solutions is two-dimensional over the field of quasi-constants. We get a basis for this space explicitly. In terms of this basis, we represent the q-hypergeometric function of the Barnes type constructed by Nishizawa and Ueno. Then we see that this function has logarithmic singularity at the origin. This is a difference between the q-hypergeometric functions wit…
Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；Laboratory of Theoretical Physics, JINR（约飞联合核子研究所理论物理实验室）；Yerevan Physics Institute（埃里安物理研究所）；National Research University Higher School of Economics（国立研究大学高等经济大学）
