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INEQUALITIES FOR THE PARTITION FUNCTION ARISING FROM TRUNCATED THETA SERIES

. Positivity questions related to the partition function arising from classical theta identities have been studied in the combinatorial and q -series framework. Two such identities that emerge from truncation of Euler’s pentagonal number theorem and an identity due to Gauss are the predominant ones among others. In this paper, we prove the asymptotic growth of coeﬃcients of truncation of theta series directly from inequalities for the shifted partition function rather than taking a detour to Wright’s circle method. Recently, Andrews and Merca conjectured that for n odd or k even, where M ( n = ( 1) 1) . We conﬁrm the conjecture for all n ≥ N ( k ) with explicit information about N ( k ) by determining the asymptotic growth of the diﬀerence between the alternating sums presented in the above inequality. This in turn shows that the conjecture of Andrews and Merca is even true for the excluded case; i.e., n even and k odd with n > N ( k ). Moreover we modify the error bound in the asymptotic expansion of M k ( n ), obtained by Chern. We also present an uniﬁed structure to obtain asymptotic growths up to any order as we please for such alternating sums involving the partition function.