Rogers-Ramanujan-Gordon Type Theorems For Overpartition

In q-series theory, the integer partition theory is a very important part, by thiscombinatorial tool named partition we can give proofs of many difficult hypergeo-metric identities intuitively. The theory of partitions has a very long history, certainspecial problems in partitions date back to the Middle Ages. But the first discoveriesof any depth were made by L. Euler, a Swiss mathematician, in the eighteenth cen-tury. Euler indeed laid the foundations of the theory of partitions. From then on, manymathematicians–Caley, Gauss, Hardy, Jacobi, Lagrange, Ramanujan, and Schur, madegreat contributions to the development of the theory of partitions.In the theory of partitions, many theorems are concerned about the equality be-tween the number of two types of partitions, that is to say, for any integer n the numbersof partitions of n are equal under the two types of restrictions. The famous Euler par-tition theorem establishes the equality between the number of partitions of n with oddparts and the number of partitions of n with distinct parts. One of the most famous the-orems of this type is the Rogers-Ramanujan theorem which states the equality betweenthe number of partitions of n with consecutive parts differ at least2and the numberof partitions of n with parts not equal to0,±2when modulo5. The theorem of thistype which states the equality between the number of partitions of n with parts satis-fying certain residue conditions and the number of partitions of n with parts satisfyingcertain difference conditions are called Rogers-Ramanujan type theorem.In1961, Gordon gave a combinatorial generalization of the Rogers-Ramanujantheorem which generalizes the modulo5in the former Roger-Ramanujan theorem toall odd integers greater than1and provided an involution proof. After that, in1966, An-drews gave an analytic proof of the Rogers-Ramanujan-Gordon theorem and in1974,Andrews also gave an analytic generalization of the Rogers-Ramanujan identity, whichcan be seen as the generating function form of the Rogers-Ramanujan-Gordon theoremand named by Andrews-Gordon identity. In1979and1980, Bressoud generalized theGordon’s theorem and Andrews’ identity to all moduli, that is to say, he gave a theorem of Rogers-Ramanujan-Gordon type with even modulo and its generating function formwhich is an identity of Andrews-Gordon type with even moduli.An overpartition of a positive integer n is a non-increasing sequence of positive in-tegers whose sum is n and the first occurrence of each integer may be overlined. By em-ploying overpartitions, we can interpret many hypergeometric series, and prove manyq-series identity naturally. Many partition theorems possess the analogues in overpar-tition theory. The main results in this paper are two overpartition analogues of theRogers-Ramanujan-Gordon theorem, and two overpartition analogues of the Andrews-Gordon identity.In Chapter2, we shall give two overpartition theorems. One of them states theequality between the number of overpartitions such that nonoverlined1occurs at mosti1times, the parts equal to l and nonoverlined parts equal to l+1occur at most k1times and the number of overpartitions of n with nonoverlined parts not congruent to0,±i when modulo2k. This theorem can be seen as the overpartition analogue of theRogers-Ramanujan-Gordon theorem. In2004, Lovejoy gave two theorems which canbe seen as the special cases of our theorem with i=1and i=k. But we can not getour theorem from his proof. The second theorem states the number of overpartitionssuch that nonoverlined1occurs at most i1times, the parts equal to l and nonoverlinedparts equal to l+1occur at most k1times and if the parts equal to l and nonoverlinedparts equal to l+1occur exactly k1times the sum of these parts are congruent to i1plus the number of overlined parts less to l+1when modulo2equals the number ofoverpartitions of n with nonoverlined parts not congruent to0,±i when modulo2k1.This theorem can be seen as the overpartition analogue of Bressoud’s theorem.In Chapter3, we shall define the Gordon marking for overpartition. It is an as-signment of marks to the parts of overpartitions such that equal parts are marked bydifferent integers and satisfy some other restrictions. For any overpartition the mark-ing is unique. In2009, Kurs ungo¨z defined the Gordon marking for ordinary partition,and by two inversion operations on the Gordon marked partition he gave a constructiveproof of the Andrews-Gordon identity. We shall also give two pairs of mutually inverseoperations on the Gordon marked overpartitions which shall play a key role in the proofof the theorems in Chapter4. In Chapter4, we shall give two Andrews-Gordon type identities which can beseen as the overpartition analogues of the Andrews-Gordon identity and Bressoud’sgeneralization of Rogers-Ramanujan identity. The two identities can also be seen asthe generating function form of the two theorems in Chapter2. We shall give theconstructive proofs of the two identities. These proofs are based on the operationsintroduced in Chapter3.