arXiv subjects

Heng Huat Chan

Publications and source records attributed to Heng Huat Chan.

7 recordsLinked to original sources

arXiv2026

On Schultz's generalization of Borweins' cubic identity

In 1991, the Borweins established a cubic analogue of Jacobi's identity for theta functions, which is used by B.C. Berndt, S. Bhargava, and F.G. Garvan in the development of Ramanujan's cubic theory of elliptic functions. In 2013, D. Schultz discovered an identity for theta series in three variables which generalizes the Borweins' identity. In this article, we revisit Schultz's identity and present two distinct approaches to its derivation. Our investigation not only provides new proofs but also yields several new Schultz-type identities.

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arXiv2021

Modular Forms and $k$-colored Generalized Frobenius Partitions

Let $k$ and $n$ be positive integers. Let $cϕ_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$ and $\mathrm{C}Φ_k(q)$ be the generating function of $cϕ_{k}(n)$. In this article, we study $\mathrm{C}Φ_k(q)$ using the theory of modular forms and discover new surprising properties of $\mathrm{C}Φ_k(q)$.

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arXiv2021

Multiplicative functions arising from the study of mutually unbiased bases

We embed the somewhat unusual multiplicative function, which was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics, into two families of multiplicative functions that we construct as generalizations of that particular example. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.

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arXiv2018

Wronskians of theta functions and series for $1/π$

In this article, we define functions analogous to Ramanujan's function $f(n)$ defined in his famous paper "Modular equations and approximations to $π$". We then use these new functions to study Ramanujan's series for $1/π$ associated with the classical, cubic and quartic bases.

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arXiv2018

Fractional Powers of the Generating Function for the Partition Function

Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$. In this article, we discuss congruences satisfied by $p_{k}(n)$ when $k$ is a rational number.

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arXiv2009

Supercongruences satisfied by coefficients of 2F1 hypergeometric series

Recently, Chan, Cooper and Sica conjectured two congruences for coefficients of classical 2F1 hypergeometric series which also arise from power series expansions of modular forms in terms of modular functions. We prove these two congruences using combinatorial properties of the coefficients.

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arXiv2004

Recent progress in the study of representations of integers as sums of squares

In this article, we collect the recent results concerning the representations of integers as sums of an even number of squares that are inspired by conjectures of Kac and Wakimoto. We start with a sketch of Milne's proof of two of these conjectures. We also show an alternative route to deduce these two conjectures from Milne's determinant formulas for sums of $4s^2$, respectively $4s(s+1)$, triangular numbers. This approach is inspired by Zagier's proof of the Kac--Wakimoto formulas via modular forms. We end the survey with recent conjectures of the first author and Chua.

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