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广义Euler数的同余及相关恒等式
【作者】 赵子盈；
【导师】 徐哲峰；
【作者基本信息】 西北大学 ， 基础数学， 2015， 硕士
【摘要】 Euler数是从组合数学中提出来的,它与著名的Fibonacci数,Bernoulli数,中心阶乘数之间有着密切的关系,因此关于Euler数的研究倍受国内外学者的关注.张文鹏,刘国栋,孙智宏等人得到了许多关于Euler数的恒等式与同余式.本文在前人研究的基础上对广义Euler数进行进一步研究.本文研究的主要内容如下：首先,总结前人关于Euler数的各种推广形式及其相关性质,在此基础上得到关于Euler数各种推广形式之间的一些恒等式.Euler数的递推公式、同余性质及反转公式.最后,结合广义k阶Euler数的定义方式,相应的给出广义k阶Euler多用对比系数法、构造法以及幂级数展开式等方法得到广义Euler多项式的一些相关恒等式.
【Abstract】 Euler number comes from combination of mathematics. It has a close re-lationship among Fibonacci numbers, Bernoulli numbers, and Central factorial numbers.Therefore, scholars both at home and aboard research on it. A lot of identity and congruence about Euler number are got by scholars, such as Zhang Wenpeng, Liu Guodong, Sun Zhihong. Based on previous studies,we have fur-ther research on the general form of the Euler numbers.The main results are presented as follows:First of all, summary definitions and basic properties about variety form of Euler numbers. On the basis of their results, we got many identities between different form of Euler numbers.Secondly, for given real numbers a, b, c, using this formula elementary methods to obtain several explicit formulas, some congruences and an inversion formula for generalizations of higher-order Euler numbers. polynomials use of contrast coefficient method and power series ex-pansion method, structure method, we got some identities of generalizations of higher-order Euler polynomials.