On monotonicity of ranks of partitions, positivity of truncated series, and transformation formulas for theta series - Ho, Thi Phuong Nhi (2017)
We discuss three different topics in combinatorial number theory. In the first topic, we form several truncated series from the quintuple product identity and its specialised versions, then prove that the coeffcients of these series exhibit uniformity in sign. In other words, all the coefficients are either positive or negative. In the second topic, we are interested in N(m,n), which denotes the number of partitions of n with rank m. Our numerical computations suggest that for m ≥ 0, n ≥ 39 and n ≠ m+ 2, we have N(m,n) ≥ N(m+ 1, n). We analyse this conjectural monotonicity and presents several useful results which could help solving it. Finally, in the last topic, we study many special eta quotients such that, when written as series, their coeffcients are interlinked in a specific manner which could be generalised. By observing that these series are associated with the quadratic form x^2 + ky^2, we construct a system of transformation formulas to prove this phenomenon of interlinked coefficients.