Series Polynomial Arithmetic In A Class Of Nature

As is well known, the Bernoulli and Fibonacci numbers have many appli-cations in many different areas of mathematics such as number theory, matric theory, combinatorics, special function and analysis. Since these two numbers appeared, the arithmetic properties of them have been investigated extensively and thoroughly by mathematicians. In recent years, some researchers found that some arithmetic identities involving the Bernoulli and Fibonacci numbers play important roles in studying some classical congruent topics and solving some conjectures. In view of the above, the arithmetic identities of some numbers and polynomials associated with the Bernoulli and Fibonacci numbers are es-tablished by using some elementary methods and techniques in this thesis. It turns out that some known results at present are obtained in brief ways and as special cases, respectively. The main achievements of this thesis are given as follows:1. By introducing and studying a new sequence of polynomials, two sym-metric identities between such a sequence of polynomials and sums of powers are established. As applications, some symmetric identities for the generalized Bernoulli polynomials of Leopoldt, the higher-order Eulcrian polynomials, the higher-order Apostol-Bernoulli and Apostol-Euler polynomials are given, and the generalized multiplication theorems of the Bernoulli and Euler polynomials due to Carlitz and some known results related to the Namias identity are obtained as special cases.2. By studying a sequence of polynomials introduced by Kejian Wu, Zhiwei Sun and Hao Pan, two symmetric identities involving such a sequence of poly-nomials are established, and some known results connected with the Kaneko idntity and the Gelfand identity are derived as special cases.3. By studying the generating functions of the Bernoulli and Euler polyno-mials, three symmetric identities involving the Bernoulli and Euler polynomials are established, by virtue of which some surprising arithmetic identities for the Bernoulli and Euler polynomials are deduced and several famous identities such as the Nielsen identity, the Miki identity, the Woodcock identity, the Matiyase-vich identity and the Zagier identity are gotten as special cases. Further, a mean value formula for products of Dirichlct L-functions is given corresponding.4. By using the above methods to obtain three symmetric identities for the Bernoulli and Euler polynomials, three quadratic recurrence formulae involving the Bernoulli and Euler polynomials are established. It turns out that the Agoh-Dilcher identity and the three multiply formulae for the Bernoulli and Euler polynomials are obtained as special cases.5. Based on the ideas stemming from the three multiply formulae for the Bernoulli and Euler polynomials given by us and the combinatorial theorem about sets called the principle of cross-classification, the Lucas sequences intro-duced by Muskat are studied, four multiply formulae involving the Lucas se-quences are established. As applications, some classical combinatorial identities for the Fibonacci and Lucas numbers are deduced as special cases.