Dissection of the quintuple product, with applications

Authors

Tim Huber, The University of Texas Rio Grande ValleyFollow
James McLaughlin, West Chester University of Pennsylvania
Dongxi Ye, Sun Yat-sen UniversityFollow

Document Type

Article

Publication Date

4-2026

Abstract

This work considers the m-dissection (for m≢0(mod3)">m≢0(mod3)) of the general quintuple productQ(z,q)=(z,q/z,q;q)∞(qz2,q/z2;q2)∞.">Q(z,q)=(z,q/z,q;q)∞(qz2,q/z2;q2)∞.Multiple novel applications arise from this m-dissection. For example, we derive the general partition identityDS(mn+(m2−1)/24)=(−1)(m+1)/6bm(n), for all n≥0,">DS(mn+(m2−1)/24)=(−1)(m+1)/6bm(n), for all n≥0,where m≡5(mod6)">m≡5(mod6) is a square-free positive integer relatively prime to 6; DS(n)">DS(n) is defined, for S the set of positive integers containing no multiples of m, to be the number of partitions of n into an even number of distinct parts from S minus the number of partitions of n into an odd number of distinct parts from S; and bm(n)">bm(n) denotes the number of m-regular partitions of n. The dissections allow us to prove a conjecture of Hirschhorn concerning the 2n">2n-dissection of (q;q)∞">(q;q)∞, as well as determine the pattern of the sign changes of the coefficients an">an of the infinite product(q2k−1;q2k−1)∞(qp;qp)∞2=∑n=0∞anqn,k≥1,p≥5a prime.">(q2k−1;q2k−1)∞(qp;qp)2∞=∞∑n=0anqn,k≥1,p≥5a prime.This covers a recent result of Bringmann et al. that corresponds to the case k=1">k=1 and p=5">p=5.

This work considers the m-dissection (for m≢0(mod3)">m≢0(mod3)) of the general quintuple productQ(z,q)=(z,q/z,q;q)∞(qz2,q/z2;q2)∞.">Q(z,q)=(z,q/z,q;q)∞(qz2,q/z2;q2)∞.Multiple novel applications arise from this m-dissection. For example, we derive the general partition identityDS(mn+(m2−1)/24)=(−1)(m+1)/6bm(n), for all n≥0,">DS(mn+(m2−1)/24)=(−1)(m+1)/6bm(n), for all n≥0,where m≡5(mod6)">m≡5(mod6) is a square-free positive integer relatively prime to 6; DS(n)">DS(n) is defined, for S the set of positive integers containing no multiples of m, to be the number of partitions of n into an even number of distinct parts from S minus the number of partitions of n into an odd number of distinct parts from S; and bm(n)">bm(n) denotes the number of m-regular partitions of n. The dissections allow us to prove a conjecture of Hirschhorn concerning the 2n">2n-dissection of (q;q)∞">(q;q)∞, as well as determine the pattern of the sign changes of the coefficients an">an of the infinite product(q2k−1;q2k−1)∞(qp;qp)∞2=∑n=0∞anqn,k≥1,p≥5a prime.">(q2k−1;q2k−1)∞(qp;qp)2∞=∞∑n=0anqn,k≥1,p≥5a prime.This covers a recent result of Bringmann et al. that corresponds to the case k=1">k=1 and p=5">p=5.

Comments

Original published version available at https://doi.org/10.1016/j.jcta.2025.106122

Recommended Citation

Huber, Tim, James Mc Laughlin, and Dongxi Ye. "Dissection of the quintuple product, with applications." Journal of Combinatorial Theory, Series A 219 (2026): 106122. https://doi.org/10.1016/j.jcta.2025.106122

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Publication Title

Journal of Combinatorial Theory, Series A

DOI

10.1016/j.jcta.2025.106122