Symbolic And Computation Method In The Study Of π-Series And Log-concavity

In this thesis,we main study the symbolic computation method in πseries and log-concavity.The study of π series originated from Remanjuan and has been widely developed later.The unimodality problem is an important research topic in combinatorics.It includes the study of unimodality,log-concavity,logconvexity,Polya frequency property etc.In the first section,we briefly introduce the background of π series and unimodality,we also introduce our main methods and results.In the second section,We investigate some hypergeometric series involving π given by Sun from the view of Gosper’s algorithm.Given a hypergeometric term tk,we consider the Gosper summability of r(k)tk for r(k)being a rational function of k.We give an upper bound and a lower bound on the degree of the numerator of r(k)such that r(k)tk is Gosper summable.We also show that the denominator of the r(k)can read from the Gosper representation of tk+1/tk.Based on these results,we give a systematic method to construct series whose sums can be derived from the known ones.We also illustrated the corresponding super-congruences and the q-analogue of the approach.In the third section,we first introduce the Zeilberger’s algorithm which is a powerful tool on finding the recurrence relations for P-recursive sequences.We then introduce the asymptotic properties of P-recursive sequences and the symbolic computation method we used.Then we treat the P-recursive sequence as the solution of the linear difference equation.We consider hypergeometric solutions of the linear difference equations and the decomposition properties of the corresponding operators.Then in the forth section,we consider the log-concavity of the combinatorial sequences.We consider the higher order Turán inequality and higher order log-concavity for sequences {an}n≥0 such that where m is a nonnegative integer,αi are real numbers,ri(x)are rational functions of x and 0<α1<α2<…<am<β.We will give a sufficient condition on the higher order Turán inequality and the l-log-concavity for n sufficiently large.Many P-recursive sequences fall in this frame.Then we will give a method to find the N such that for any n>N,the higher order Turan inequality holds.Many examples such as the partition function are shown in this section.