Composition-theoretic series and false theta functions (Q6551226)

Partition-theoretic series can be interpreted in terms of composition-theoretic series. In this paper, the authors emphasize this point. To elucidate this connection, several basic theorems from [\textit{S. Radu} and \textit{J. A. Sellers}, Int. J. Number Theory 7, No. 8, 2249--2259 (2011; Zbl 1246.11168)] and [\textit{R. Schneider} and \textit{A.V. Sills}, Ramanujan J. 65, No. 4,1863--1881 (2024; Zbl 07951624)] relating to partition-theoretic and composition-theoretic are reviewed. Consequently, the authors study the false theta function: \N\[\N\Psi(a,b):= \sum_{n=0}^{\infty}a^{ \binom{n+1}{2}}b^{\binom{n}{2} }- \sum^{-1}_{n=-\infty} a^{ \binom{n+1}{2}}b^{\binom{n}{2}}.\N\]\NThis function leads to several related theorems. In particular, two significant theorems are as follows:\N\NTheorem 1. For\N\[\N\sum_{n=0}^{\infty} c(n) q^n =\frac{1}{\left(q\right)_{\infty}-2q^5},\N\]\Nthe coefficients satisfy \(c(n) >F_n\), the \(n\)-th Fibonacci number.\N\NTheorem 2. Let \(1 \leq a < b < c\), \(a, b, c\in \mathbb{N}\). Then the coefficients of \(\frac{1-q^a- q^b+q^c}{\left(q\right)_{\infty}}=\sum_{n=0}^{\infty} (p(n)-p(n-a)-p(n-b)+p(n-c))q^n\) are positive if \(c \leq a+b\), and eventually at least some if not all are negative of increasing absolute value if \(c > a+ b\).\N\NThe second theorem extends the assertion regarding the eventual negativity of the expression \[p(n)- p(n -1) -p(n -2) + p(n -5)\] from \textit{G. E. Andrews} and \textit{M. Merca}'s result in [J. Comb. Theory, Ser. A 119, No. 8, 1639--1643 (2012; Zbl 1246.05014)]. Some questions and conjectures are also proposed, for example:\N\NConjecture 1. Let \( a > 0 \) and \( j \in \mathbb{N} \).\N\begin{itemize}\N\item[(1)] The coefficients of the function\N\[\N\frac{1}{(q)_{\infty} - aq^j} = \frac{1}{(q)_{\infty} \left(1 - \frac{aq^j}{(q)_{\infty}}\right)}\N\]\Ngrow exponentially.\N\item[(2)] The coefficients of the function\N\[\N\frac{1}{(q)_{\infty} + aq^j} = \frac{1}{(q)_{\infty} \left(1 + \frac{aq^j}{(q)_{\infty}}\right)}\N\]\Nalternate in sign, with the maximum absolute value of the first \( n \) terms growing exponentially.\N\end{itemize}\N\NConjecture 2. The coefficients of \( H(q)\) in \(\frac{1}{\Psi(-q^2,q)}=\frac{1}{(q)_{\infty}(1-H(q)) } \) are strictly positive. The coefficients of \(\frac{1}{\Psi(-q^2,q)}\) grow exponentially.