Dongxi Ye

Dongxi Ye
Sun Yat-Sen University | SYSU · School of Mathematics (Zhuhai)

Doctor of Philosophy

About

36
Publications
3,669
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171
Citations
Citations since 2016
25 Research Items
153 Citations
20162017201820192020202120220102030
20162017201820192020202120220102030
20162017201820192020202120220102030
20162017201820192020202120220102030

Publications

Publications (36)
Article
Full-text available
In recent work, Z.-H. Sun proposes a large number of conjectural formulas on interrelations between representations by a sum of triangular numbers and representations by ternary or quaternary quadratic forms. In this note, we establish a general statement to illustrate that all these conjectural formulas can be rigorously verified with the aid of m...
Article
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In this work, we verify all the conjectural formulas for the Mahler measure of the Laurent polynomial $$ \left(X+\frac{1}{X}\right)^{2}\left(Y+\frac{1}{Y}\right)^{2}(1+Z)^{3}Z^{-2}-s $$ parametrized by $s$ posed by Samart using properties of spherical theta functions, and show that when $s$ is induced by a CM point, these Mahler measures are all ex...
Article
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In this note, we revisit Ramanujan’s function k(τ) and discuss various results, old and new, on it in modular-functional context.
Article
In this work, we show that the difference of a Hauptmodul for a genus zero group Γ0(N) as a Hilbert modular function on Y0(N) × Y0(N) is a Borcherds lift of type (2, 2). As applications, we derive Monster denominator formula like product expansions for these Hilbert modular functions and certain Gross-Zagier type CM value formulas.
Article
In this note, we confirm a conjectural formula on the number of representations of the square of an odd prime by a sum of an odd number of squares, i.e., $$ \#\{(x_{1},\ldots,x_{2k+1}\in\mathbb{Z}^{2k+1}|\,x_{1}^{2}+\cdots+x_{2k+1}^{2}=p^{2}\}, $$ proposed by Cooper.
Article
Klein forms are used to construct generators for the graded algebra of modular forms of level 7. Dissection formulas for the series imply Ramanujan type congruences modulo powers of 7 for a family of generating functions that subsume the counting function for 7-core partitions. The broad class of arithmetic functions considered here enumerate color...
Article
In \cite{CY}, Chen and Yui conjectured that Gross--Zagier type formulas may also exist for Thompson series. In this work, we verify Chen and Yui's conjecture for the cases for Thompson series $j_{p}(\tau)$ for $\Gamma_{0}(p)$ for $p$ prime, and equivalently, establish formulas for the prime decomposition of the resultants of two ring class polynomi...
Article
In recent work, we use Dudek’s method together with a result of Zagier to establish an asymptotic formula for the average number of divisors of an irreducible quadratic polynomial of the form \(x^{2}-bx+c\) with b, c integers. In this note, we remark that one can adopt the work of Hooley to derive a more precise asymptotic formula for the case \(x^...
Article
In this work, we establish Ramanujan–Mordell type formulas associated to certain quadratic forms of discriminants 20^k or 32^k. In the end, a remark is given to illustrate potential extensions to other related cases.
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We derive a formula for the discriminant of the ring class polynomial of conductor N associated to the j ‐invariant and an imaginary quadratic field of odd discriminant − d using the theory of Borcherds lifts. When d is a prime and N = 1 , the formula recovers the Gross–Zagier discriminant formula.
Article
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Two level 17 modular functions r=q2∏n=1∞(1-qn)n17,s=q2∏n=1∞(1-q17n)3(1-qn)3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} r = q^2 \prod _{n=1}^{\infty...
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In this work, we employ a well known result due to Zagier to derive an asymptotic formula for the average number of divisors of a quadratic polynomial of the form \(x^{2}-bx+c\) with \(b,\,c\) integers. As simple consequences, we obtain formulas for computing the narrow class number of a quadratic field and relations between the narrow class number...
Article
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For a positive integer $N$, we derive a Kronecker type limit formula for a Dedekind type zeta function $\zeta_{\mathcal{O}_{K}(N)}(s;[\mathfrak{a}])$ associated to a wide ideal class of a quadratic order of conductor $N$ of a quadratic field $K$. As consequences, we establish a Chowla--Selberg type formula for a modular form for $\Gamma_{0}(N)$, gi...
Article
In this note, we use automorphic Green functions to show that the generating function of a canonical basis for the space of weakly holomorphic modular functions with poles supported at the cusp \(i\infty \) for a Fuchsian subgroup of the first kind of genus zero has a weight 2 meromorphic modular form representation.
Article
In this work, we study various properties of two families of normal subgroups of the Hecke group Γ0(3) and their associated modular curves and function fields. As consequences, we obtain the stabilizer subgroups of the N-th roots of a uniformizer for X(Γ0(3)), and realize such N-th roots as uniformizers for the associated modular curves to their st...
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In this work, we derive Gross–Zagier type CM value formulas for Hauptmoduls jp⁎(τ) on Fricke groups Γ0⁎(p). We also illustrate how to employ these formulas to obtain certain Hilbert class polynomials.
Article
In this short note, we realize a product associated to a uniformizer for genus zero \(\Gamma _{0}(N)\) twisted with a genus character as a twisted Borcherds product lifted from a weight \(\frac{1}{2}\) harmonic weak Maass form for the Weil representation.
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In this note, we construct canonical bases for the spaces of weakly holomorphic modular forms with poles supported at the cusp ∞ for Γ 0(4 ) of integral weight k for k≤ - 1 , and we make use of the basis elements for the case k= - 1 to construct explicit Borcherds products on unitary group U (2 , 1 ).
Article
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Properties of theta functions and Eisenstein series dating to Jacobi and Ramanujan are used to deduce differential equations associated with McKay Thompson series of level 20. These equations induce expansions for modular forms of level 20 in terms of modular functions.The theory of singular values is applied to derive expansions for $1/\pi$ of lev...
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In this short note, we aim to prove that the Fourier coefficients of the modular function $ \frac{\eta^{24}(\tau)}{\eta^{24}(2\tau)} $ possess a sign-change property.
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In this note, we construct canonical bases for the spaces of weakly holomorphic modular forms with poles supported at the cusp $\infty$ for $\Gamma_{0}(4)$ of integral weight $k$ for $k\leq-1$, and we make use of the basis elements for the case $k=-1$ to construct explicit Borcherds products on unitary group $U(2,1)$.
Article
The Ramanujan--Mordell Theorem for sums of an even number of squares is extended to other quadratic forms and quadratic polynomials.
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The analogous theory for level 16 of Ramanujan's theories of elliptic functions to alternative bases is developed by studying the level 16 modular function
Article
We provide a comprehensive study of the function $h=h(q)$ defined by $$\begin{eqnarray}h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})}\end{eqnarray}$$ and show that it has many properties that are analogues of corresponding results for Ramanujan’s function $k=k(q)$ defined by $$\begin{eqnarray...
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For any positive integer n, we state and prove a formula for the number of solutions in integers of n = x2 + xy + 7y2 + z2 + zt + 7t2.
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In this work, we establish formulas for the representations of binary quadratic forms of discriminants -44, -92, -108, -135 and -140 as a sum of Lambert series and eta-quotients.
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We briefly review Ramanujan's theories of elliptic functions to alternative bases, describe their analogues for levels 5 and 7, and develop new theories for levels 14 and 15. This gives rise to a rich interplay between theta functions, eta-products and Eisenstein series. Transformation formulas of degrees five and seven for hypergeometric functions...
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In this work, we evaluate the convolutions ∑l+36m=n σ(l)σ(m) and ∑4l+9m=n σ(l)σ(m) for all positive n. As an application, these evaluations are used to determine the number of representations of a positive integer by the forms and .
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We employ a modular method to establish the new result that two types of Eisenstein series to the tredecic base may be parametrised in terms of the eta quotients \${\it\eta}^{13}({\it\tau})/{\it\eta}(13{\it\tau})\$ and \${\it\eta}^{2}(13{\it\tau})/{\it\eta}^{2}({\it\tau})\$. The method can also be used to give short and simple proofs for the analog...
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The theory of theta functions is used to derive hypergeometric transformation formulas of degrees 3, 7, 11 and 23. As a consequence of the theory that is developed, some new series for are obtained that are similar to a class investigated by Ramanujan.
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The theory of quasimodular forms is used to evaluate the convolution sums for all positive integers n. As a consequence, the number of representations of a positive integer n by the octonary quadratic form is determined.
Article
The Rogers–Ramanujan continued fraction has a representa-tion as an infinite product given by q1/5∏j=1∞(1−qj)(j5) |q|<1|q|<1 (jp) q q∏j=1∞(1−qj)(j13).
Article
One of the properties of the Rogers-Ramanujan continued fraction is its representation as an infinite product given by where (frac(j, p)) is the Legendre symbol. In this work we study the level 13 function R (q) = q underover(∏, j = 1, ∞) (1 - qj)(frac(j, 13)) and establish many properties analogous to those for the fifth power of the Rogers-Ramanu...
Article
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For any positive integer n and for certain fixed positive integers a 1 ,a 2 ,⋯,a 7 , the authors study the number of solutions in integers of a 1 x 1 2 +a 2 x 2 2 +a 3 x 3 2 +a 4 x 4 2 +a 5 x 5 2 +a 6 x 6 2 +a 7 x 7 2 =n 2 When a 1 =⋯=a 7 =1, this reduces to the classical formula for the number of representations of a square as a sum of seven squar...

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