Combinatorial Identities And Generating Function Method

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.