Harmonic Numbers And Its Infinite Series Identities

Harmonic numbers is an important research object in combinatorial mathematics and special function theory.It has been widely used in number theory,computer algebra,theoretical physics,computer biology and other fields.Discovering and proving combinatorial identities with harmonic numbers is one of the hot topics for scholars nowadays.Applying Abel's lemma on summation by parts,we consider infinite series identities with harmonic numbers and summation expressions about special functions in this paper.The contents is summirized as follows:The introduction illustrates related concepts and development background of harmonic numbers,the development status at home and abroad.In chapter 2,we give some basic definitions about generalized harmonic numbers,Riemann Zeta function,Abel's lemma on summation by parts,partial fractional decomposition and list several important summation formulas.In chapter 3,based on Abel's lemma on summation by parts,we study summation of infinite series involving generalized harmonic numbers and obtain some infinite series related to harmonic numbers.Furthermore,we give several summation expressions of irrational number ?,logarithm In 2.In chapter 4,applying Abel's lemm on summation by parts,we establish summation of alternative series including generalized harmonic numbers,furthermore we get the alternative series identities related to harmonic numbers and derive some summation expressions of irrational number ?,logarithm ln 2,Catalan constant in the meantime.