两类Cauchy数的共同推广

使用包含两个参数的一般阶乘，第一类和第二类Cauchy数被统一为广义Cauchy数．对该数的指数型生成函数，得到了它的封闭形式，利用广义Cauchy数的定义和它的生成函数导出该数的两个递推关系．广义Cauchy数和广义Stirling数之间的一个变换公式显示它们之间的密切联系，运用积分的计算技巧，证明了广义Cauchy数卷积和广义Stirling数之间的一个关系。最后．用Bell多项式和第二类Bernoulli数分别给出了广义Cauchy数的两种不同表示。

Generalization of two kinds of Cauchy numbers

Cauchy numbers of the first and the second kind can be unified into the generalized Cauchy numbers by starting with transformations between generalized factorials involving two arbitrary parameters. A closed form for the exponential generating function of the generalized Cauchy numbers is obtained. Two recurrence relations for generalized Cauchy numbers are also deduced by their definition and generating function, respectively. A transform formula between the generalized Cauchy numbers and the generalized Stirling numbers indicates their close relation. The convolution relation between the generalized Cauchy numbers and the generalized Stirling numbers is shown by using intergral skills. Finally, the generalized Cauchy numbers are expressed by the Bell polynomials and the second kind Bernoulli numbers, respectively.