Asymptotic Enumeration Methods And Their Applications

In this thesis, we give asymptotics of special combinatorial sums, the number of alignmentsof biosequences and inverse moment of positive binomial and negative binomial by asymptoticenumeration methods.The main contents of this thesis can be summarized as follows:In Chapter 2, we give precise formulae of special combinatorial sums by generating functionand further study the asymptotics of combinatorial sums sum from k=0 to n ((?))~Ï„k~l and sum from k=1 to n ((?))~Ï„1/k~l.In recent years, the studies on the identities involving the inverse of binomial coefficientshave gradually increased, but not many studies on the asymptotic behaviors of the summationof the inverse of binomial coefficients. In Chapter 3, we get some of results of finite sumsand infinite series involving powers and inverse of binomial coefficients by means of generatingfunctions and the integral theory. We also make use of asymptotic enumeration methods tocompute the asymptotic values.In Chapter 4, we extend Rice's Lemma and study the asymptotics of the alternative com-binatorial sums by Rice's Lemma and residue theorem and give its q- analog.In Chapter 5, we give precise formulae of the two-dimensional recursion sequences by gen-erating function and study its asymptotics by asymptotic enumeration methods.In Chapter 6, we give precise formulae of the number of alignments of biosequences, andmake use of generating function asymptotic estimation technique to compute its asymptoticvalues.In Chapter 7, we give asymptotic expansions of the first inverse moment of positive binomialand negative binomial by singularity analysis method, and further study the asymptotics for r-thinverse moment of positive binomial and negative binomial in terms of the poissonization anddepoissonization method.