Congruence Property Of Partition Functions Based On Theta Function Identities

The theory of partititons is one of the most active fields,and it has numerous applications in many fields,such as "Bin Packing" and "Currency Exchange".This is exactly the reason why many well-known scholars and researchers,including Professor George E.Andrews a member of Academy of Sciences,who are interested in it.This thesis focuses on the properties of overpartition functions,t-regular partition functions,3-component multipartitions,bipartitions with distinct even parts and t-core partition functions based on the theory of theta function identities.These partition functions have numerous natural connections to many branches of mathematics,such as Mathematical Physics;the representation theory of Algebra and Combinatorics.We investigate the arithmetic properties of these partition functions and establish infinite families of congruences for these partition functions.Furthermore,we study the properties of coefficients of Fourier expansions for several eta quotients and generalize some results due to Proessor Kenneth S.Williams.This thesis is organized as follows.In chapter 1 and 2,we introduce basic background of the theory of partitions and the theory of theta function identities.Moreover,we introduce some properties for overpartition functions,t-regular partition functions,k-component multipartitions,bipartitions with distinct even parts and t-core partition functions.The above theory were proved by a number of scholars and researchers.In chapter 3,we establish new theta function identities and present new dissection formula for overpartition functions.From the dissection formula,we derive 5 new congruences for ovarpartition functions which solve an open problem given by Professor Michael D.Hirschhorn.In chapter 4,we give a characterization on the parity results for 11-,13-and 17-regular partition functions based on theta function identities and the theory of quadratic residue.Furthermore,we obtain several infinite families of congruences modulo 2 for tregular partition functions,which generalize some results due to Professor James A.Sellers.In chapter 5,we investigate the arithmetric properties for 3-component multipartitions and then prove a number of congruences modulo powers of 3 by utilizing theta function identities.Our results generalize some congruences for 3-component multipartitions which were proved by Professor Nayandeep D.Baruah.In chapter 6,we study the arithmetric properties for bipartitions with distinct even parts.Dai proved some congruences modulo 2,4 and 8 for bipartitions with distinct even parts.We establish some 2-and 3-dissection formulas for certain theta functions and then employ these formulas to prove several infinite families of congruences modulo 16,32 and 64 for bipartitions with distinct even parts.Therefore,our results generalize the congruences due to Dai.In chapter 7,we present a characterization on the parity results for 15-core partition functions by using the theory of modular equations.First,we establish the relations between the theory of modular equations and the theory of theta functions and deduce some theta function identities from some modular equations.Next,we employ these theta function identities to give a characterization on the parity results for 15-core partition and discover several infinite families of congruences modulo 2 for 15-core partition functions.In chapter 8,we investigate the properties of Fourier coefficients for certain eta quotients.We establish the recurrence relations of Fourier coefficients for certain eta quotients by employing theta function identities.We give certain eta quotients such that their Fourier coefficients vanish for all positive integers in each of infinitely many nonoverlapping arithmetic progressions.We not only give a new proof of Williams' results,but also generalize his results.