On the positivity of truncated pentagonal number series and some conjectures of Merca
Abstract
Let ν2(m) denote the $2$-adic valuation of a positive integer m and set Nm=m(1+ν2(m)/2). We prove four conjectures of Merca on the nonnegativity of truncated pentagonal number series weighted by the infinite products ∏m≥ 1(1−q2Nm) and ∏r≥ 1(q2r r;q2r+1r)∞. Our method is to regard the exponent map m↦ m(ν2(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−q4)(1−q5)). 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.
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BibTeX
@misc{saikia2026positivity,
title = {{On the positivity of truncated pentagonal number series and some conjectures of Merca}},
author = {Manjil P. Saikia and Abhishek Sarma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25739},
eprint = {2609.25739},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25739},
note = {CC0},
}
RIS
TY - UNPB
TI - On the positivity of truncated pentagonal number series and some conjectures of Merca
AU - Manjil P. Saikia
AU - Abhishek Sarma
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25739
UR - https://arxiv.org/abs/2609.25739
AB - Let ν_2(m) denote the $2$-adic valuation of a positive integer m and set N_m=m(1+ν_2(m)/2). We prove four conjectures of Merca on the nonnegativity of truncated pentagonal number series weighted by the infinite products ∏_m≥ 1(1−q^2N_m) and ∏_r≥ 1(q^2^r r;q^2^r+1r)_∞. Our method is to regard the exponent map m↦ m(ν_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.
ER -