Concerning a General Source Formula and Its Applications

DOI：10.3770/j.issn:2095-2651.2016.05.001

 作者 单位 徐利治 大连理工大学数学科学学院, 辽宁 大连 116024

本文是对不久前发表的基本成果[见Anal. Theory Appl. (分析学理论及应用), 2015, 31(3): 260-282]作出推进和发展.证明前文中提出的普遍本源公式还包括了作者与合作者(何天晓、薛昭雄)早前所建立的一对级数变换公式作为其特例.这样一来,可知普遍本源公式实际上已蕴含有50多个特殊公式及恒等式作为其推论[参见Disc. Math. (离散数学杂志), 2008, 308: 3427-3440]. 由普遍本源公式能导出的一切公式构成一个公式类,称之为\$\Sigma\Delta D\$类.这个公式类至少包括有20来个有名的经典公式作为其成员.又因普遍本源公式及其所属三个本源公式所引出的`` 嵌入技巧''还可用以继续寻求新而有趣的特殊公式,因此\$\Sigma\Delta D\$类应是离散数学分析中最便于应用且能不断觅得有趣新成员的公式类.

Here presented is a further investigation on a general source formula (GSF) that has been proved capable of deducing more than 30 special formulas for series expansions and summations in the author's recent paper [On a pair of operator series expansions implying a variety of summation formulas. Anal. Theory Appl., 2015, 31(3): 260--282]. It is shown that the pair of series transformation formulas found and utilized by He, Hsu and Shiue [cf. Disc. Math., 2008, 308: 3427--3440] is also deducible from the GSF as consequences. Thus it is found that the GSF actually implies more than 50 special series expansions and summation formulas. Finally, several expository remarks relating to the \$(\Sigma\Delta D)\$ formula class are given in the closing section.