Bernoulli Polynomials And Power Sums

Bernoulli numbers and polynomials have numerous important applications in number theory, combinatorics, and classical numerical analysis. Bernoulli polynomials have been studied extensively over the last two centuries. In the research related to this subject, the identities involving such numbers and polynomials catch plenty of interests to many authors. The key point of this thesis is a part of progress in deriving new identities involving Bernoulli and Euler polynomials and we give an extension to the classical Faulhaber's theorem related to Bernoulli polynomials.In Chapter 1, the basic properties of classical Bernoulli and Euler numbers and polynomials are introduced. Then we show some different generalizations of Bernoulli polynomials which will be used in later chapters. One important theorem, Faulhaber's theorem, which is related to powers of sums is also mentioned.At the beginning of Chapter 2, we present several known Bernoulli identities involving both ordinary convolution and binomial convolution on Bernoulli polynomials. Then we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. We use this generalized formula to derive a symmetric identity which can be used to reduce to some known identities on Bernoulli polynomials and Bernoulli numbers. Finally, two generalizations on the symmetric Bernoulli identities are obtained.Another approach to derive Bernoulli polynomials are studied in Chapter 3. By using the classical inverse relation of Stirling numbers, we give two inverse pairs of identities involving products of the Bernoulli polynomials and the Bernoulli polynomials of the second kind.In Chapter 4, we show that the classical Faulhaber's theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a + b, a + 2b,..., a + nb is a polynomial in na + n(n + 1)b/2. The coefficients of these polynomials are given in terms of the Bernoulli polynomials. Following Knuth's approach by using the central factorial numbers, we obtain formulas for r-fold sums of powers. Moreover, we consider the r-fold alternating sum of powers, and give an explicit formula in terms of Euler polynomials.In the last chapter, we proceed with the power sums of binomial polynomials. By using a multiple transformation identity of hypergeometric series, we show that power sums of (?) has a factor (?). Let m be the power of each summands, when rm is odd and r≥3, the summation has a factor (?)~2 . A new proof of Schmidt's problem is also obtained by the transformation identity.