The Rogers-Ramanujan-Gordon Theorem for Overpartitions

Details The Rogers-Ramanujan-Gordon Theorem for Overpartitions The Rogers-Ramanujan-Gordon Theorem for Overpartitions

2011-08-30

arxiv.org/abs/1108.5792v1

Mathematics

Combinatorics

26 pages

Scientific paper

Let $B_{k,i}(n)$ be the number of partitions of $n$ with certain difference condition and let $A_{k,i}(n)$ be the number of partitions of $n$ with certain congruence condition. The Rogers-Ramanujan-Gordon theorem states that $B_{k,i}(n)=A_{k,i}(n)$. Lovejoy obtained an overpartition analogue of the Rogers-Ramanujan-Gordon theorem for the cases $i=1$ and $i=k$. We find an overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general case. Let $D_{k,i}(n)$ be the number of overpartitions of $n$ satisfying certain difference condition and $C_{k,i}(n)$ be the number of overpartitions of $n$ whose non-overlined parts satisfy certain congruences condition. We show that $C_{k,i}(n)=D_{k,i}(n)$. By using a function introduced by Andrews, we obtain a recurrence relation which implies that the generating function of $D_{k,i}(n)$ equals the generating function of $C_{k,i}(n)$. We also find a generating function formula of $D_{k,i}(n)$ by using Gordon marking representations of overpartitions, which can be considered as an overpartition analogue of an identity of Andrews for ordinary partitions.

Affiliated with

Also associated with

No associations

Rating

The Rogers-Ramanujan-Gordon Theorem for Overpartitions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The Rogers-Ramanujan-Gordon Theorem for Overpartitions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Rogers-Ramanujan-Gordon Theorem for Overpartitions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-728683