arXiv subjects

Eric H. Liu

Publications and source records attributed to Eric H. Liu.

5 recordsLinked to original sources

arXiv2026

A proof of Andrews-El Bachraoui's conjecture on the parity of coefficients of a $q$-series

Recently, Andrews and El Bachraoui studied a partition function $s_1(n)$, which counts the number of two-color partitions into distinct parts of $n$ whose smallest part occurs in one prescribed color only, while every larger part may occur in either color or in both colors. They obtained a complete description modulo 4 for $s_1(n)$. They also considered a $q$-series $T_{o}(q)$ which is the odd companion series of the generating function for $s_1(n)$. At the end of their paper, they presented a conjecture on the parity of the coefficients of $T_o(q)$. In this paper, we confirm this conjecture. Moreover, we establish an infinite family of congruences modulo 8 for the coefficients of $S_1(q)$ and prove that the set of integers satisfying $s_1(n)\equiv 0\pmod 8$ has natural density one.

math.NT↗

arXiv2026

Some identities on the second order mock theta functions

Recently, Nath and Das investigated congruence properties for the second order mock theta function $B(q)$. In their paper, they asked for analytic proofs of three identities on the second order mock theta functions $A(q)$, $B(q)$ and $μ_2(q)$. In this paper, we settle Nath and Das' open problem by using the $(p, k)$-parametrization of theta functions and several identities due to Hickerson and Mortenson.

math.NT↗

arXiv2022

Proofs of some conjectures of Chan-Mao-Osburn on Beck's partition statistics

Recently, George Beck introduced two partition statistics $NT(m,j,n)$ and $M_ω(m,j,n)$, which denote the total number of parts in the partition of $n$ with rank congruent to $m$ modulo $j$ and the total number of ones in the partition of $n$ with crank congruent to $m$ modulo $j$, respectively. Andrews proved a congruence on $NT(m,5,n)$ which was conjectured by Beck. Very recently, Chan, Mao and Osburn established a number of Andrews-Beck type congruences and posed several conjectures involving $NT(m,j,n)$ and $M_ω(m,j,n)$. Some of those conjectures were proved by Chern and Mao. In this paper, we confirm the remainder three conjectures of Chan-Mao-Osburn and two conjectures due to Mao. We also present two new conjectures on $M_ω(m,j,n)$ and $NT(m,j,n)$.

math.CO↗

arXiv2010

Partition Identities for Ramanujan's Third Order Mock Theta Functions

We find two involutions on partitions that lead to partition identities for Ramanujan's third order mock theta functions $ϕ(-q)$ and $ψ(-q)$. We also give an involution for Fine's partition identity on the mock theta function f(q). The two classical identities of Ramanujan on third order mock theta functions are consequences of these partition identities. Our combinatorial constructions also apply to Andrews' generalizations of Ramanujan's identities.

math.CO↗

arXiv2009

A Franklin Type Involution for Squares

We find an involution as a combinatorial proof of a Ramanujan's partial theta identity. Based on this involution, we obtain a Franklin type involution for squares in the sense that the classical Franklin involution provides a combinatorial interpretation of Euler's pentagonal number theorem. This Franklin type involution can be considered as a solution to a problem proposed by Pak concerning the parity of the number of partitions of n into distinct parts with the smallest part being odd. Using a weighted form of our involution, we give a combinatorial proof of a weighted partition theorem derived by Alladi from Ramanujan's partial theta identity. This answers a question of Berndt, Kim and Yee. Furthermore, through a different weight assignment, we find combinatorial interpretations for another partition theorem derived by Alladi from a partial theta identity of Andrews. Moreover, we obtain a partition theorem based on Andrews' identity and provide a combinatorial proof by certain weight assignment for our involution. A specialization of our partition theorem is relate to an identity of Andrews concerning partitions into distinct nonnegative parts with the smallest part being even. Finally, we give a more general form of our partition theorem which in return corresponds to a generalization of Andrews' identity.

math.CO↗