Abstract

An investigation of applied problems related to heat conduction and dynamics, electromagnetic oscillations and aerodynamics, quantum mechanics and potential theory leads to the study of various hypergeometric functions. The great success of the theory of hypergeometric functions in one variable has stimulated the development of corresponding theory in two and more variables. In the theory of hypergeometric functions, an increase in a number of variables will always be accompanied by a complication in the study of the function of several variables. Therefore, the decomposition formulas that allow us to represent the hypergeometric function of several variables through an infinite sum of products of several hypergeometric functions in one variable are very important, and this, in turn, facilitate the process of studying the properties of multidimensional functions. In the literature, hypergeometric functions are divided into two types: complete and confluent. In all respects, confluent hypergeometric functions including the decomposition formulas, have been little studied in comparison with other types of hypergeometric functions, especially when the dimension of the variables exceeds two. In this paper we define a new class of confluent hypergeometric functions of several variables, study their properties and determine the system of hypergeometric equations that these functions satisfy, because all fundamental solutions of the generalized Helmholtz equation with singular coefficients are written out through one new introduced confluent hypergeometric function of several variables. Using the decomposition formulas which are established here the order of the singularity of the found fundamental solutions of the elliptic equation which mentioned above is determined.

中文翻译：

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具有几个奇异系数和多个变量合一超几何函数的广义亥姆霍兹方程的基本解

摘要

对与热传导和动力学，电磁振荡和空气动力学，量子力学和势能理论有关的应用问题的研究导致了对各种超几何函数的研究。一个变量的超几何函数理论的巨大成功刺激了两个或更多变量的相应理论的发展。在超几何函数理论中，许多变量的增加总是伴随着对若干变量函数的研究的复杂化。因此，允许我们通过一个变量中多个超几何函数的乘积的无穷大来表示多个变量的超几何函数的分解公式非常重要，因此，有助于研究多维函数属性的过程。在文献中，超几何函数分为两种：完全和融合。在所有方面，与其他类型的超几何函数相比，包括分解公式在内的融合超几何函数很少得到研究，尤其是当变量的维数超过2时。在本文中，我们定义了一类新的融合多个变量的超几何函数，研究了它们的性质并确定了这些函数满足的超几何方程组，因为通过一个新的引言写出了具有奇异系数的广义亥姆霍兹方程的所有基本解融合的几个变量的超几何函数。