Related papers: Extending the Meijer $G$-function

We present a new definition of Euler Gamma function. From the complex analysis and transalgebraic viewpoint, it is a natural characterization in the space of finite order meromorphic functions. We show how the classical theory and formulas…

The paper is devoted to the piece-wise analytic case of Meijer's $G$ function $G^{m,n}_{p,p}$. While the problem of its analytic continuation was solved in principle by Meijer and Braaksma we show that in the ''balanced'' case $m+n=p$ the…

We study $K$-theoretic integrals over famed quiver moduli via wall-crossing phenomena. We study the chainsaw quiver varieties, and consider generating functions defined by two types of $K$-theoretic classes. In particular, we focus on…

By a similar idea for the construction of Milnor's gamma functions, we introduce "higher depth determinants" of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant…

Recently, there emerges different versions of beta function and hypergeometric functions containing extra parameters. Gaining enlightenment from these ideas, we will first introduce a new extension of generalized hypergeometric function and…

Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a…

For compact sets $K\subset \mathbb C^{d}$, we introduce a subalgebra $A_{D}(K)$ of $A(K)$, which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.

The umbral approach provides methods for comprehending and redefining special functions. This approach is employed efficiently in order to uncover intricacies and introduce new families of special functions. In this article, the umbral…

Recently, Shehata et al. [37] introduced the $_{r+1}R_{s,k}(B,C,z)$ matrix function and established some properties. The aim of this study established to devote and derive certain basic properties including analytic properties, recurrence…

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.

In our recent work we proposed a generalization of the beta integral method for derivation of the hypergeometric identities which can by analogy be termed "the G function integral method". In this paper we apply this technique to the cubic…

A new generalization of the modified Bessel function of the second kind $K_{z}(x)$ is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby…

The A-hypergeometric system studied by I.M. Gelfand, M.I. Graev, A.V. Zelevinsky and the author, is defined for a set A of characters of an algebraic torus. In this paper we propose a generalization of the theory where the torus is replaced…

The multiple gamma function $\Gamma_n$, 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…

We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating…

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…

The authors survey recent results in special functions, particularly the gamma function and the Gaussian hypergeometric function.

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…

One of the goals of the present paper is to propose an elementary method to find a general formula for the Fourier transform containing a pair of complex gamma functions with a monomial sm in terms of Gauss's hypergeometric functions 2F1.…

We open a new perspective on the sup-norm problem and propose a version for non-spherical Maass forms when the maximal compact K is non-abelian and the dimension of the K-type gets large. We solve this problem for an arithmetic quotient of…