微分方程$y^{(k)}=e^{ay}$的解、Bell多项式的特殊值和$(k,a)$ -自治系数
Solution of the Differential Equation $y^{(k)}=e^{ay}$, Special Values of Bell Polynomials and $(k,a)$-Autonomous Coefficients
摘要：
本文利用方程$y^{(k)}=e^{ay}$的幂级数解给出了Bell多项式的特殊值。此外，还定义了完全和部分指数自治函数、指数自治多项式、自治多项式和$(k,a)$ -自治系数.最后，我们给出了组合对象的各种计数与二项式系数、第二类Stirling数和自治系数之间的关系。
英文摘要：
In this paper special values of Bell polynomials are given by using the power series solution of the equation $y^{(k)}=e^{ay}$. In addition, complete and partial exponential autonomous functions, exponential autonomous polynomials, autonomous polynomials and $(k,a)$-autonomous coefficients are defined. Finally, we show the relationship between various numbers counting combinatorial objects and binomial coefficients, Stirling numbers of the second kind and autonomous coefficients.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
