7 research outputs found
Two truncated identities of Gauss
Two new expansions for partial sums of Gauss' triangular and square numbers
series are given. As a consequence, we derive a family of inequalities for the
overpartition function and for the partition function
counting the partitions of with distinct odd parts. Some further
inequalities for variations of partition function are proposed as conjectures.Comment: 9 pages, final versio
Expansions of averaged truncations of basic hypergeometric series
Motivated by recent work of George Andrews and Mircea Merca on the expansion
of the quotient of the truncation of Euler's pentagonal number series by the
complete series, we provide similar expansion results for averages involving
truncations of selected, more general, basic hypergeometric series. In
particular, our expansions include new results for averaged truncations of the
series appearing in the Jacobi triple product identity, the -Gau{\ss}
summation, and the very-well-poised summation. We show how special
cases of our expansions can be used to recover various existing results. In
addition, we establish new inequalities, such as one for a refinement of the
number of partitions into three different colors.Comment: 14 p