On the Definition of Higher Gamma Functions

IF 2.3 2区 数学 Q1 MATHEMATICS
Ricardo Pérez-Marco
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引用次数: 0

Abstract

We extent our definition of Euler Gamma function to higher Gamma functions, and we give a unified characterization of Barnes higher Gamma functions, Mellin Gamma functions, Barnes multiple Gamma functions, Jackson q-Gamma function, and Nishizawa higher q-Gamma functions in the space of finite order meromorphic functions. The method extends to more general functional equations and unveils the multiplicative group structure of solutions that appears as a cocycle equation. We also generalize Barnes hierarchy of higher Gamma function and multiple Gamma functions. With the new definition, Barnes–Hurwitz zeta functions are no longer necessary in the definition of Barnes multiple Gamma functions. This simplifies the classical definition, without the analytic preliminaries about the meromorphic extension of Barnes–Hurwitz zeta functions, and defines a larger class of Gamma functions. For some algebraic independence conditions on the parameters, we prove uniqueness of the solutions. Hence, this implies the identification of classical Barnes multiple Gamma functions as a subclass of our multiple Gamma functions.

论高伽马函数的定义
我们将欧拉伽马函数的定义扩展到了高伽马函数,并给出了有限阶微观函数空间中巴恩斯高伽马函数、梅林伽马函数、巴恩斯多重伽马函数、杰克逊q-伽马函数和西泽高q-伽马函数的统一表征。该方法可扩展到更一般的函数方程,并揭示了作为循环方程出现的解的乘法组结构。我们还概括了高伽马函数和多重伽马函数的巴恩斯层次结构。根据新定义,巴恩斯多重伽马函数的定义中不再需要巴恩斯-赫尔维茨zeta函数。这就简化了经典定义,省去了关于巴恩斯-赫尔维茨zeta函数的分形扩展的分析预处理,并定义了更大类的伽马函数。对于参数的某些代数独立条件,我们证明了解的唯一性。因此,这意味着经典巴恩斯多重伽马函数是我们的多重伽马函数的一个子类。
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来源期刊
CiteScore
3.50
自引率
3.70%
发文量
35
审稿时长
1 months
期刊介绍: Constructive Approximation is an international mathematics journal dedicated to Approximations and Expansions and related research in computation, function theory, functional analysis, interpolation spaces and interpolation of operators, numerical analysis, space of functions, special functions, and applications.
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