Browsing by Author "Chan, Heng Huat"

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A Cubic Analogue of the Jacobsthal Identity
(2011-04-01)
Chan, Heng Huat
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Long, Ling
;
Yang, YiFan
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應用數學系
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Department of Applied Mathematics
It is well known that if p is a prime such that p I (mod 4). then p can be expressed as a sum of two squares. Several proofs of this fact are known and one of them, due to E. Jacobsthal, involves the identity p = x(2) + y(2) with x and y expressed explicitly in terms of sums involving the Legendre symbol. These sums are now known as the Jacobsthal sums. In this short note, we prove that if p equivalent to 1 (mod 6). then 3p = u(2) + uv + nu(2) for some integers u and v using an analogue of Jacobsthal's identity.
CONGRUENCES MODULO 5 AND 7 FOR 4-COLORED GENERALIZED FROBENIUS PARTITIONS
(2017-10-01)
Chan, Heng Huat
;
Wang, Liuquan
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Yang, Yifan
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應用數學系
;
Department of Applied Mathematics
Let c phi(k)(n) denote the number of k-colored generalized Frobenius partitions of n. Recently, new Ramanujan-type congruences associated with c phi(4)(n) were discovered. In this article, we discuss two approaches in proving such congruences using the theory of modular forms. Our methods allow us to prove congruences such as c phi(4)(14n + 6) 0 mod 7 and Seller's congruence c phi(4)(10n + 6) 0 mod 5.
New analogues of Clausen's identities arising from the theory of modular forms
(2011-10-01)
Chan, Heng Huat
;
Tanigawa, Yoshio
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Yang, Yifan
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Zudilin, Wadim
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應用數學系
Around 1828, T. Clausen discovered that the square of certain hypergeometric (2)F(1) function can be expressed as a hypergeometric (3)F(2) function. Special cases of Clausen's identities were later used by S. Ramanujan in his derivation of 17 series for 1/pi. Since then, there were several attempts to find new analogues of Clausen's identities with the hope to derive new classes of series for 1/pi. Unfortunately, none were successful. In this article, we will present three new analogues of Clausen's identities. Their discovery is motivated by the study of relations between modular forms of weight 2 and modular functions associated with modular groups of genus 0. (C) 2011 Elsevier Inc. All rights reserved.