On the integration of the differential equations of five‐parametric double‐hypergeometric functions of second order
On the integration of the differential equations of five‐parametric double‐hypergeometric functions of second order
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二阶五参数双超几何函数微分方程的积分
DOI：
10.1063/1.523405
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发表时间：
1977
影响因子：
1.3
通讯作者：
P. Olsson
中科院分区：
文献类型：
--
作者：
P. Olsson
将appels超几何函数F2 （a,b1,b2,c1,c2; x1,x2）的微分方程视为两个合流超几何函数乘积的拉普拉斯变换，得到其解。由于这些微分方程可以转化为与Appell函数F3（a,b1,b2,c1,c2; x1,x2）和Horn函数H2（a,b,c,d,e; x1,x2）相关的方程，因此这些方程可以同时求解。给出了六种级数的36个不同解的集合。这个集合是最小的集合，它解释了三个双超几何二阶函数F2， F3和H2中的任何一个的一般行为。给出了用级数和积分表示的解的各种形式，以及解之间的联系，这些解可以解析地延续函数。
Solutions of the differential equations associated with Appell’s hypergeometric function F2 (a,b1,b2,c1,c2; x1,x2) are obtained by considering them as a Laplace transform of a product of two confluent hypergeometric functions. Since these differential equations may be transformed into the equations associated with Appell’s function F3(a,b1,b2,c1,c2; x1,x2) and Horn’s function H2(a,b,c,d,e; x1,x2), these equations are solved simultaneously. A set of 36 distinct solutions in terms of six types of series is given. This set is the smallest set which accounts for the general behavior of any of the three double‐hypergeometric second order functions F2, F3, and H2. Various representations of the solutions in terms of series and integrals are given as well as connections between the solutions which continue the functions analytically.