On asymptotic formulas for certain $q$-series involving partial theta functions

Abstract

In this article, we investigate the integer sequence a d (n), where for a positive integer d, a d (n) is defined byIn particular, we will present a combinatorial model for a d (n) via integer partitions and this will play a crucial role in obtaining an asymptotic formula for a d (n). Moreover, we will show that p(n) − 2a d (n) has an unexpected sign pattern via combinatorial arguments and asymptotic formulas.

On the Andrews–Warnaar identities for partial theta functions

“…For asymptotic expansions, we refer the reader to B.C. Berndt and B. Kim [10], S. Jo and B. Kim [18], B. Kim and J. Lovejoy [22], R. R. Mao [33], R. J. McIntosh [34], B. Kim, E. Kim, and J. Seo [21]. Especially noteworthy is that in [18], Jo and Kim defined the integer sequence {a n (d)} n≥0 by the expansion…”

Partial theta function identities from Wang and Ma's conjecture

“…For more formulas related to partial theta functions, the reader may refer to the papers [7,9,10,11].…”

On the Subpartitions of the Ordinary Partitions, II

“…In [9], the author uses subpartitions to find combinatorial proofs of entries in Ramanujan's lost notebook [11]. Moreover, these subpartitions play a crucial role in obtaining an asymptotic formula for certain q-series involving partial theta functions [8].…”

“…Since n copies of 1 is always counted by p e (n, a, (a + b)/2), the positivity of q-expansion is now clear from the combinatorial description. The case a = b = 1 appears Andrews [2] as a generating function for the number of partitions of n in which the first non-occurrence number as a part is odd, which is a conjugation of partition with subpartition of even length with gap d as noted in [8]. When a = 1 and b = 3, we have Moreover, by adopting the argument in [8], we can prove the following theorem.…”

“…Remark 6. The case a = 2 and b = 0 was discussed in [8,Theorem 2]. In this case, the sign of p e (n, 2, 1) − p o (n, 2, 1) is alternating.…”