On the irregular cases of the linear ordinary difference equation
On the irregular cases of the linear ordinary difference equation
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DOI：
10.1090/s0002-9947-1928-1501443-6
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发表时间：
1928-03
影响因子：
1.3
通讯作者：
C. Raymond;C. R. A. July
中科院分区：
文献类型：
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作者：
C. Raymond;C. R. A. July
通常假设(3)的根是有限的、不同的且不等于零。这种狭义的限制性假设描述了我们所说的常规情况。但是很少有人注意到更普遍的不规则情况的问题，在这种情况下，根不能满足这个假设。Barnesf（1905）用拉普拉斯变换研究了系数为x线性的二阶方程的大多数不规则情形。1910年，Horn|给出了形式级数解（发散的），并给出了一个定理，该定理证明了x的大实正值的渐近形式是由形式级数给出的，当唯一偏离正则性的是一个根的消失时。在1913年学会之前读过的论文（按标题）中，Batchelder§发现了秒方程在所有可能的不规则情况下的形式级数解
the assumption is commonly made that the roots of (3) are finite, distinct, and different from zero. This narrowly restrictive hypothesis characterizes what we call the regular case. But little attention has yet been devoted to the more general problem of the irregular cases, in which the roots fail to satisfy this hypothesis. Barnesf in 1905 studied most of the irregular cases of the equation of second order with coefficients linear in x, employing the Laplace transformation. In 1910 Horn| gave formal series solutions (divergent) and a theorem setting forth the existence of solutions whose asymptotic forms for large real positive values of x are given by the formal series, when the only departure from regularity is the vanishing of one root. In papers read (by title) before the Society in 1913 Batchelder§ found formal series solutions in all possible irregular cases for the equation of second