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Zhi-Wei Sun

Publications and source records attributed to Zhi-Wei Sun.

At least 37 records · Page 2Linked to original sources
arXiv2026

Series involving central binomial coefficients and higher-order harmonic numbers

We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to reduce the corresponding series to special values of the Dirichlet $L$-functions. For example, we establish the following identities conjectured by Sun:\[\sum_{k=0}^\infty\binom{2k}{k}^3\left[ \mathsf H_{2k}^{(2)}-\frac{25}{92}\mathsf H_{ k}^{(2)} +\frac{735L_{-7}(2)-86π^{2}}{1104}\right]\frac{1}{4096^{k}}=0,\]\[\sum_{k=0}^\infty\binom{2k}k^3\left[\mathsf H_{2k}^{(3)}-\frac{43}{352}\mathsf H_k^{(3)}\right]\frac{42k+5}{4096^k}=\frac{555ζ(3)}{77π}-\frac{32G}{11},\] where $ \mathsf H^{(r)}_k:= \sum_{0<n\leq k}\frac{1}{n^r}$, $ L_{-7}(2):= \sum_{n=1}^\infty\left(\frac{-7}{n}\right)\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^{2}}-\frac{1}{5^{2}}-\frac{1}{6^{2}}+\frac{1}{8^{2}}+\cdots $, $ G:= \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^2}$, and $ ζ(3):= \sum_{n=1}^\infty\frac1{n^3}$.

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arXiv2026

Universal sums via products of Ramanujan's theta functions

An integer-valued polynomial $P(x,y,z)$ is said to be universal (over $\mathbb Z$) if each nonnegative integer can be written as $P(x,y,z)$ with $x,y,z\in\mathbb Z$. In this paper, we mainly introduce a new technique to determine the universality of some sums in the form $x(a_1x+a_2)/2+y(b_1y+b_2)/2+z(c_1z+c_2)/2$ (with $a_1-a_2,b_1-b_2,c_1-c_2$ all even) conjectured by Sun, using various identities of Ramanujan's theta functions. For example, we prove that $x(3x+1)+y(3y+2)+2z(3z+2)$ and $x(4x+r)+y(3y+1)/2+z(7z+1)/2\ (r=1,3)$ are universal.

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arXiv2026

Taylor coefficients and series involving harmonic numbers

During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $58$ series identities involving harmonic numbers, eight of which were previously conjectured by the second author. For example, we obtain that \[ \sum_{k=1}^{\infty} \frac{(-1)^k}{k^2{2k \choose k}{3k \choose k}} \left( \frac{7 k-2}{2 k-1} H_{k-1}^{(2)}-\frac{3}{4 k^2} \right) = \frac{π^4}{720}. \] and \[ \sum_{k=1}^\infty \frac{1}{k^2 {2k \choose k}^2} \left( \frac{30k-11}{k(2k-1)} (H_{2k-1}^{(3)} + 2 H_{k-1}^{(3)}) + \frac{27}{8k^4} \right) = 4 ζ(3)^2, \] where $H_n^{(m)}$ denotes $\sum_{0<j \le n}j^{-m}$.

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arXiv2026

A family of polynomials and related congruences and series

In this paper we study a family of polynomials $$S_n^{(m)}(x):=\sum_{i,j=0}^n\binom ni^m\binom nj^m\binom{i+j}ix^{i+j}\ \ (m,n=0,1,2,\ldots).$$ For example, we show that $$\sum_{k=0}^{p-1}S_k^{(0)}(x)\equiv\frac x{2x-1}\left(1+\left(\frac{1-4x^2}p\right)\right)\pmod p $$ for any odd prime $p$ and integer $x\not\equiv1/2\pmod p$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also formulate some open conjectures on related congruences and series for $1/π$. For example, we conjecture that $$\sum_{k=0}^\infty(7k+1)\frac{S_k^{(2)}(1/11)}{9^k}=\frac{5445}{104\sqrt{39}\,π}$$ and $$\sum_{k=0}^\infty(1365k+181)\frac{S_k^{(2)}(1/18)}{16^k}=\frac{1377}{\sqrt2\,π}.$$

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arXiv2026

New series involving binomial coefficients (III)

We evaluate some series with summands involving a single binomial coefficient $\binom{6k}{3k}$. For example, we prove that $$\sum_{k=0}^\infty\frac{(63k^2+78k+22)8^k}{(2k+1)(6k+1)(6k+5)\binom{6k}{3k}}=\frac{3π}2.$$ Motivated by Galois theory, we introduce the so-called Duality Principle for irrational series of Ramanujan's type or Zeilberger's type, and apply it to find 26 new irrational series identities. For example, we conjecture that \begin{align*}&\sum_{k=1}^\infty\frac{(32(91\sqrt{33}-523))^{k}}{k^3\binom{2k}k^2\binom{3k}k} \left((91\sqrt{33}+891)k-33\sqrt{33}-225\right) \\&\qquad=320\left(\frac{11}3\sqrt{33}L_{-11}(2)-27L_{-3}(2)\right), \end{align*} where $ L_{d}(2)=\sum_{k=1}^\infty\frac{(\frac{d}k)}{k^2}$ for any integer $d\equiv0,1\pmod4$ with $(\frac{d}k)$ the Kronecker symbol.

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arXiv2026

New series involving binomial coefficients (II)

In this paper, we evaluate some series of the form $$\sum_{k=1}^\infty\frac{ak^2+bk+c}{k(3k-1)(3k-2)m^k\binom{4k}k}.$$ For example, we prove that $$\sum_{k=1}^\infty\frac{(5k^2-4k+1)8^{k}}{k(3k-1)(3k-2)\binom{4k}k}=\frac{3}2π$$ and $$\sum_{k=1}^\infty\frac{415k^2-343k+62}{k(3k-1)(3k-2)(-8)^k\binom{4k}k}=-3\log2.$$ We also pose many new conjectural series identities involving binomial coefficients; for example, we conjecture that $$\sum_{k=0}^\infty\frac{\binom{2k}k^3}{4096^k}\left(9(42k+5)\sum_{0\le j<k}\frac1{(2j+1)^4}+\frac{25}{(2k+1)^3}\right)=\frac 56π^3.$$

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arXiv2026

On series identities involving $\binom{4k}k$ and harmonic numbers

The harmonic numbers are those $H_n=\sum_{0<k\le n}\frac1k\ (n=0,1,2,\ldots)$. In this paper we confirm over ten conjectural series identities with summands involving the binomial coefficient $\binom{4k}k$ and harmonic numbers. For example, we prove the identities $$\sum_{k=1}^\infty \frac{\binom{4k}{k}}{16^k}\left((22k^2-92k+11)H_{4k}-\frac{449k-275}{2}-\frac{85}{12k}\right)=-151-\frac{80}{3}\log{2}$$ and $$ \sum_{k=0}^\infty\frac{\binom{4k}{k}((11k^2+8k+1)(10H_{4k}-17H_{2k})+2k+18)}{(3k+1)(3k+2)16^k}=8\log2,$$ which were previously conjectured by Z.-W. Sun.

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arXiv2025

Undecidability on Diophantine equations over $\mathbb Z[i]$ with $20$ unknowns

It is known that Hilbert's Tenth Problem over the Gaussian ring $\mathbb Z[i]=\{a+bi:\ a,b\in\mathbb Z\}$ is undecidable. In this paper we obtain the following further result: There is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{20})=0$ (with integer coefficients) is solvable over $\mathbb Z[i]$. This improves the previous record involving $52$ variables.

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arXiv2025

Evaluation of a determinant involving Legendre symbols

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $A_p(x)$ denote the matrix $[x+a_{ij}]_{1\leqslant i,j\leqslant (p-1)/2}$, where $$ a_{ij}=\begin{cases} (\frac{j}{p}) &\text{if} \ i=1, \$\frac{i+j}{p}) &\text{if} \ i>1. \end{cases}$$ In 2018 Z.-W. Sun conjectured that $\det A_p(0)=-2^{(p-3)/2}$ if $p\equiv 3 \pmod{4}$, which was later confirmed by G. Zaimi. In this paper we evaluate $\det A_p(x)$ completely.

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arXiv2025

Fast converging irrational series for $ L(2,(\frac d\cdot))$

By exploring the theory of Guillera-Rogers, we evaluate some infinite series whose summands are quadratic irrationals, in terms of $π$ and special values of Dirichlet $L$-functions $ L_d(2)\equiv L(2,(\frac d\cdot)):=\sum_{k=1}^\infty\left( \frac{d}{k} \right)\frac1{k^2}$. Applying Kronecker's theorem to linear combinations of lattice sums, we obtain geometrically convergent series for $ L_{-56}(2)$, $ L_{-68}(2)$, $ L_{-87}(2)$, $ L_{-111}(2)$, and $ L_{-116}(2)$, which go beyond the solvable cases of Guillera-Rogers.

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arXiv2025

Some determinants involving binary forms

In this paper, we study arithmetic properties of certain determinants involving powers of $i^2+cij+dj^2$, where $c$ and $d$ are integers. For example, for any odd integer $n>1$ with $(\frac dn)=-1$ we prove that $\det [ (\frac{i^2+cij+dj^2}{n})]_{0\le i,j\le n-1}$ is divisible by $φ(n)^2$, where $(\frac{\cdot}{n})$ is the Jacobi symbol and $φ$ is Euler's totient function. This confirms a previous conjecture of the second author.

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arXiv2025

On $p$-adic congruences involving $\sqrt d$

Let $p$ be an odd prime and let $d$ be an integer not divisible by $p$. We prove that $$ \prod_{1\le m,n\le p-1\atop p\nmid m^2-dn^2}\ (x-(m+n\sqrt{d})) \equiv \begin{cases}\sum_{k=1}^{p-2}\frac{k(k+1)}2x^{(k-1)(p-1)}\pmod p &\text{if}\ (\frac dp)=1,\\\sum_{k=0}^{(p-1)/2}x^{2k(p-1)} \pmod p&\text {if}\ (\frac dp)=-1, \end{cases}$$ where $(\frac dp)$ denotes the Legendre symbol. This extends a recent conjecture of N. Kalinin. We also obtain the Wolstenholme-type congruence $$\sum_{1\le m,n\le p-1\atop p\nmid m^2-dn^2}\ \ \frac1{m+n\sqrt d}\equiv0\pmod{p^2}.$$

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arXiv2025

On determinants involving $(\frac{j+k}p)\pm(\frac{j-k}p)$

Let $p=2n+1$ be an odd prime. In this paper, we mainly evaluate determinants involving $(\frac {j+k}p)\pm(\frac{j-k}p)$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. When $p\equiv1\pmod4$, we determine the characteristic polynomials of the matrices $$\left[\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)\right]_{1\le j,k\le n}\ \ \text{and}\ \ \left[\left(\frac{j+k}p\right)-\left(\frac{j-k}p\right)\right]_{1\le j,k\le n},$$ and also establish the general identity \begin{align*} &\ \left|x+\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac jp\right)y+\left(\frac kp\right)z+\left(\frac{jk}p\right)w\right|_{1\le j,k\le n} \\=&\ (-p)^{(p-5)/4}\left(\left(\frac{p-1}2\right)^2wx-\left(\frac{p-1}2y-1\right)\left(\frac{p-1}2z-1\right)\right). \end{align*}

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arXiv2025

Evaluations of some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$

In this paper, via the beta function we evaluate some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$. For example, we prove that $$\sum_{k=0}^\infty\frac{(49k+1)8^k}{3^k\binom{3k}k}=81+16\sqrt3\,π\ \ \text{and}\ \ \sum_{k=0}^\infty\frac{10k-1}{\binom{4k}{2k}}=\frac{4\sqrt 3}{27}π.$$ We also establish the following efficient formula for computing $\log n$ with $1<n\le 85/4$: \begin{align*} &\sum_{k=0}^\infty\frac{(2(n^2+6n+1)^2(n^2-10n+1)k+P(n))(n-1)^{4k}} {(-n)^k(n+1)^{2k}\binom{4k}{2k}}\\ \ \ &=6n(n+1)(n-1)^3\log n-32n(n+1)^2(n^2-4n+1), \end{align*} where $$P(n):=n^6-58n^5+159n^4+52n^3+159n^2-58n+1.$$ In addition, we pose some conjectures on series whose summands involve $\binom{2k}k/(\binom{3k}k\binom{6k}{3k})\ (k\in\mathbb N)$.

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arXiv2025

Series with summands involving harmonic numbers

For each positive integer $m$, the $m$th order harmonic numbers are given by $$H_n^{(m)}=\sum_{0<k\le n}\frac1{k^m}\ \ (n=0,1,2,\ldots).$$ We discover exact values of some series involving harmonic numbers of order not exceeding four. For example, we conjecture that $$\sum_{k=0}^\infty(6k+1)\frac{\binom{2k}k^3}{256^k}\left(H_{2k}^{(3)}-\frac{7}{64}H_{k}^{(3)}\right) =\frac{25ζ(3)}{8π}-G,$$ where $G$ denotes the Catalan constant $\sum_{k=0}^\infty(-1)^k/(2k+1)^2$. This paper contains $70$ conjectures posed by the author during 2022--2023.

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arXiv2024

New results similar to Lagrange's four-square theorem

In this paper we establish some new results similar to Lagrange's four-square theorem. For example, we prove that any integer $n>1$ can be written as $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\in\mathbb Z$. Let $a$ and $b$ be integers with $a>0$, $b>-a$ and $\gcd(a,b)=1$. When $2\nmid ab$, we show that any sufficiently large integer can be written as $$\frac{w(aw+b)}2+\frac{x(ax+b)}2+\frac{y(ay+b)}2+\frac{z(az+b)}2$$ with $w,x,y,z$ nonnegative integers. When $2\mid a$ and $2\nmid b$, we prove that any sufficiently large integer can be written as $$w(aw+b)+x(ax+b)+y(ay+b)+z(az+b)$$ with $w,x,y,z$ nonnegative integers.

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arXiv2024

A new trigonometric identity with applications

In this paper we obtain a new curious identity involving trigonometric functions. Namely, for any positive odd integer $n$ we prove that $$\sum_{k=1}^n(-1)^k(\cot kx)\sin k(n-k)x=\frac{1-n}2,$$ which is equivalent to the identity $$\sum_{k=1}^n(-1)^kU_{n-k}(\cos kx)=-\frac{n+1}2,$$ where $U_m(z)$ stands for the $m$th Chebyshev polynomial of the second kind. As a consequence, for any positive odd integer $n$ and positive integer $m$ we obtain $$\sum_{k=1}^n(-1)^kk^{2m}B_{2m+1}\left(\frac{n-k}2\right)=0,$$ where $B_j(x)$ denotes the Bernoulli polynomial of degree $j$.

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arXiv2024

Permanent identities, combinatorial sequences, and permutation statistics

In this paper, we confirm six conjectures on the exact values of some permanents, relating them to the Genocchi numbers of the first and second kinds as well as the Euler numbers. For example, we prove that $$\mathrm{per}\left[\left\lfloor\frac{2j-k}{n}\right\rfloor\right]_{1\le j,k\le n}=2(2^{n+1}-1)B_{n+1},$$ where $B_0,B_1,B_2,\ldots$ are the Bernoulli numbers. We also show that $$ \mathrm{per}\left[\mathrm{sgn}\left(\cosπ\frac{i+j}{n+1}\right)\right]_{1\le i,j\le n}=\begin{cases} -\sum_{k=0}^m\binom{m}{k}E_{2k+1}&\quad\text{if}\ n=2m+1,\\ \sum_{k=0}^m\binom{m}{k}E_{2k}&\quad\text{if}\ n=2m, \end{cases} $$ where $\mathrm{sgn}(x)$ is the sign function, and $E_0,E_1,E_2,\ldots$ are the Euler (zigzag) numbers. In the course of linking the evaluation of these permanents to the aforementioned combinatorial sequences, the classical permutation statistic -- the excedance number, together with several kinds of its variants, plays a central role. Our approach features recurrence relations, bijections, as well as certain elementary operations on matrices that preserve their permanents. Moreover, our proof of the second permanent identity leads to a proof of Bala's conjectural continued fraction formula, and an unexpected permutation interpretation for the $γ$-coefficients of the $2$-Eulerian polynomials.

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