论正切数(戴煦数)的计数意义

TANGENT NUMBERS(DAIXU NUMBERS):ITS MEANING IN THE COUNTING THEORY OF COMBINATORICS

正切数(戴煦数)是1种与欧拉数相匹配的特殊函数和计数函数,作为独立的数学对象,19世纪中叶开始引起中外数学家的注意。本文着重分析了正切数在组合数学计数理论中的意义,简介西方数学家J.G regory(1671),Schlom ich(1857),Andre(1879,1881,1883,1894,1895),Estanave(1902),Schwartz(1931),Toscano(1936),Entrenger(1966)和Knuth,Buckholtz(1967)的成果。主要介绍晚清浙江数学家徐有壬(1840年前)、戴煦(1852)在中国数学史上的开创性工作和数学史界李俨(1955)等的研究。

As a special function and counting function,Dai Xu s number(tangent number) is matching with Euler s number(secant number),but as a self-existent mathematics research object,Chinese and western mathematicians pay attention to it till in the middle of 19th century.This paper intends to show,what is tangent number and its meaning in the counting theory of combinatorics.We simply introduce the study of western mathematicians,such as Schlomich(1857),Andre(1879,1881,1883,1894,1895),Estanave(1902),Schwartz(1931),...