Studies On Euler Sums Involving Powers Of Two

The Euler sums involving powers of two are special series on harmonic numbers.They are also the generalizations of classical Euler sums and alternating Euler sums.But so far,there are still a large number of Euler sums'values that have not been determined.In this thesis,using generating functions,integrals of special functions,compositions and permutations,alternating multiple zeta values and multiple polylogarithms,we study the Euler sums involving powers of two systematically.The main contents are as follows:?1?Using generating functions and integrals of special functions,we establish various Stirling sums and other identities related to Euler sums involving powers of two.More-over,we give the expressions of these Stirling sums by alternating multiple zeta values,and establish further the expressions of some special types of Euler sums.?2?Combining with the values of alternating Euler sums and the alternating multiple zeta values,we use these Stirling sums and identities to compute the Euler sums involving powers of two and of weight w?6.As a result,all the Euler sums involving powers of two and of weight w?5 are reducible to polylogarithms,ln?2?and zeta values,and all the sums of weight 6 are expressible in terms of polylogarithms,ln?2?,zeta values and???5,?1?.?3?Based on these works,using compositions and permutations,we establish the ex-pressions of Euler sums involving powers of two by multiple polylogarithms,and show that all these Euler sums are also expressible in terms of alternating multiple zeta values.By spec-ifying the parameters,we further discuss some special cases,including linear sums S_p,q?1/2?,quadratic sums S_i1i2,q?1/2?and nonlinear sums S_rm,_q?1/2?.According to the results,we de-velop a Maple program to compute these Euler sums,and obtain the values of the sums of weight 7.Finally,combining with the different expressions of some special sums,we estab-lish three identities on alternating multiple zeta values,including a conjectural one due to Borwein et al.