Asymptotic inequalities for k-ranks and their cumulation functions

作者：Mao Renrong^*

来源：Journal of Mathematical Analysis and Applications, 2014, 409(2): 729-741.

DOI：10.1016/j.jmaa.2013.07.057

• Asymptotic
• Cumulation functions
• Circle method
• k-ranks
• Inequalities
• Partial theta functions
• q-series

Asymptotic formulas for the positive moments of rank and crank of partitions were obtained by K. Bringmann and K. Mahlburg recently. Motivated by their works, in this paper, we prove asymptotic formulas for the k-ranks and their cumulation functions. Asymptotic inequalities between these combinatorial objects are also discovered. In particular, we show that, for fixed integer land sufficiently large N, M(l, N) similar to N(l, N) and M(l, N) < N(l, N),
where, M(I, N) (resp. (l, N)) denotes the number partitions of N with crank (resp. rank) l.

• 出版日期2014-1-15