On the positivity of truncated pentagonal number series and some conjectures of Merca

Abstract

Let ${\nu }_2(m)$ denote the $2$-adic valuation of a positive integer $m$ and set $N_m=m(1+{\nu }_2(m)/2)$. We prove four conjectures of Merca on the nonnegativity of truncated pentagonal number series weighted by the infinite products ${\prod }_{m\ge 1}(1-q^{2N_m})$ and ${\prod }_{r\ge 1}(q^{2^{r}r};q^{2^{r+1}r}{)}_{\infty }$. Our method is to regard the exponent map $m\mapsto m({\nu }_2(m)+2)$ as a dynamical system on the positive integers: the factors of the associated quotient link into chains along its forward orbits, and the three orbits seeded at $1$, $4$ and $5$ are pairwise disjoint and telescope to exactly $1/((1-q)(1-q^4)(1-q^5))$. The exponent triple $(1,4,5)$ is admissible in the sense of earlier work by Liu, which yields the desired factorization into two series with nonnegative coefficients. This orbit telescoping technique appears to be a mechanism complementary to the Pólya--Szeg\H o criterion that underlies most existing positivity results of this kind in the literature.

arXiv comment16 pages