Fundamental solutions of the generalized Helmholtz equation with several singular coefficients and confluent hypergeometric functions of many variables

Decomposition formulas associated with the multivariable confluent hypergeometric functions

The main object of this work is to show how some rather elementary techniques based upon certain inverse pairs of symbolic operators would lead us easily to several decomposition formulas associated with confluent hypergeometric functions of two and more variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities several decomposition formulas are found, which express the aforementioned hypergeometric functions in terms of such simpler functions as the products of the Gauss hypergeometric function…

Complementary Polynomials From Rodrigues' Representations For Confluent And Hypergeometric Functions And More

Complementary polynomials of Legendre polynomials are briefly presented, as well as those for the confluent and hypergeometric functions, relativistic Hermite polynomials and corresponding new pre-Laguerre polynomials. The generating functions are all given in closed form and are much simpler than the standard ones. Some are simply polynomials in two variables. New recursions and addition formulas are derived.

与三变量超几何函数相关的偏微分方程组及其在原点附近的特解

PDE-Systems associated with the hypergeometric functions in three variables and their particular solutions near the origin

本文系统构造了 600 个三变量合流超几何函数所满足的偏微分方程组，并在原点附近求出了部分方程组的特解，拓展了 Horn 的多变量超几何级数理论。

On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function of four variables. In addition, the order of the singularity is determined and the properties of the found fundamental solutions that are necessary for solving boundary value problems for degenerate elliptic equations of second order are found.

Double-Layer Potentials for a Generalized Bi-Axially Symmetric Helmholtz Equation II

The double-layer potential plays an important role in solving boundary value problems for elliptic equations. All the fundamental solutions of the generalized bi-axially symmetric Helmholtz equation were known, and only for the first one was constructed the theory of potential. Here, in this paper, we aim at constructing theory of double-layer potentials corresponding to the next fundamental solution. By using some properties of one of Appell's hypergeometric functions in two variables, we prove limiting theorems and derive integral equations concerning a denseness of double-layer potentials.

Euler and Laplace integral representations of GKZ hypergeometric functions

We introduce an interpolation between Euler integral and Laplace integral: Euler-Laplace integral. We establish a combinatorial method of constructing a basis of the rapid decay homology group associated to Euler-Laplace integral with a nice intersection property. This construction yields a remarkable expansion formula of cohomology intersection numbers in terms of GKZ hypergeometric series. As an application, we obtain closed formulas of the quadratic relations of Aomoto-Gelfand hypergeometric functions and their confluent analogue in terms of bipartite graphs.