【作者】 赵凤珍；

【导师】 王天明；

【作者基本信息】 大连理工大学 ， 计算数学， 2004， 博士

【摘要】 本文利用发生函数理论建立了一系列新的组合恒等式，并且利用发生函数方法讨论了一些级数的取整值问题以及某些多项式在有理点的取值问题。本文主要工作可概括如下： 1．第二章：到目前为止，涉及二项式系数的倒数的恒等式并不多。本章利用定积分的方法和常见的发生函数，结合一些运算技巧，建立了一些与二项式系数的倒数有关的恒等式。特别，我们建立的恒等式推广了人们熟知的结果 2．第三章：我们利用发生函数及其它知识讨论Rogers-Ramanujan型恒等式问题： (1)Rogers-Ramanujan型恒等式在众多学科中的广泛应用，促使人们研究如何发现并证明这类恒等式。关于这类恒等式的研究一直是非常活跃的问题，并产生了许多证明方法(见文献[18，22-23，29，41，56，70-71，78])。在文献[41]中，Gessel与Stanton利用Lagrange反演的q-模拟推导出一个优美的Rogers-Ramanujan型恒等式本章将这个恒等式的证明归结为如何计算级数(其中s为整数)，给出了计算该级数的递推公式。此外还从另一角度推广了上面提到的恒等式。 (2)我们对Agarwal与Singh(见文献[1])推导出的一个新的Rogers-Ramanujan型恒等式给出了一个新的、更为简单的证法，并得出一系列新的Roger*Ramanujan型恒等式. (3)本章借助一个基本超几何函数的变换公式、已知的恒等式以及Jacobi三重积，建立了一些新的Rogers-Ramanujan型恒等式. 3.第四章:利用发生函数理论以及一些运算技巧，讨论与广义Fi-bonaeei，Lueas数有关的间题: (l)利用发生函数原理，建立了一系列涉及广义Fibonacci，Lucas数的幂的恒等式，并由这些恒等式得到了新的同余关系. (z)与广义Fibonacci，Lucas数的倒数有关的级数的计算问题，难度很大，很难给出精确值.在本章第二节中，我们将Lambert级数法与Theta函数法结合起来，计算出一些与广义Fibonacci，Lucas数的倒数有关的级数的近似值. (3)利用发生函数理论和Peu方程的有关知识，我们解决了下列级数在有理点的取整值间题:及(x)一息知>0)，su*(x)一叠知:0)，sV，k(x)一叠知七。其中{巩}，{玖}分别为广义Fibonaeei，Lueas序列. 4.第五章:由于Berno过一i多项式与Hurwitz’5 zeta函数之间存在紧密联系，因此Bernoulli多项式与Eule:多项式在有理点的取值间题引起了人们的关注(见文献!3，38，74一饲).本章利用发生函数原理研究与Bernouili数、E过e:数关系密切的高阶Eule:多项式、Genoe面多项式、sali‘多项式、N6rlund Euze:多项式以及N6rlund Bernoulh多项式在有理点的取值问题.更多还原

【Abstract】 In this thesis, by employing generating function technique, some new combinatorial identities have been established. Furthermore, the integral values of some kinds of series involving generalized Fibonacci and Lucas numbers at rational numbers are given and the values of some polynomials at rational points have been studied as well. The main results of the thesis can be summarized as follows:1. In Chapter 2, by means of the integral theory, a number of reciprocal series involving binomial coefficients have been established and some results related to binomial coefficients have been extended. In particular, we obtain2. Rogers-Ramanujan type identities play a very important role in many fields and have been receiving much attention in the literature. We study this kind of identities in Chapter 3.(1) Making use of q-analog of Lagrange inversion, Gessel and Stanton [41] give a graceful identityWe have shown that the identity can be obtained by calculating the series 1 + We further derive the recurrence formula of computing thisseries. Moreover, the identity mentioned above has also been generalized. (2) Agarwal and Singh [1] obtained an identityWe have found a simpler proof of the above identity. In addition, we derive a number of new identities.(3) With the help of a transformation formula of basic hypergeometric functions, known identities, and Jacobi triple, we have established some new Rogers-Ramanujan type identities.3. Fibonacci and Lucas numbers are very important in many subjects such as number theory, geometry, and algebraic. We investigate this kind of numbers in Chapter 4 and give the following results:(1) By generating functions theory, we have established a number of identities involving the powers of generalized Fibonacci and Lucas numbers. From these identities, we can obtain some new congruences.(2) It is difficult to compute reciprocal series involving generalized Fibonacci and Lucas numbers. We are concerned with this problem. By combining with the methods of Theta functions and Lambert series, we compute some reciprocal series related to generalized Fibonacci and Lucas numbers.(3) By generating function theory and knowledge of Pell equations, we compute the integral values of the following series at rational numbers:where {Un} and {Vn} stand for generalized Fibonacci and Lucas sequences, respectively.4. Bernoulli polynomials and Euler polynomials are very useful in number theory and special functions. In Chapter 5, by generating function theory and skills of computation, we study the values of Euler polynomials of higher order, Genocchi polynomials, Salie polynomials, Norlund Euler polynomials and Norlund Bernoulli polynomials at rational numbers and obtain the values of these polynomials.更多还原