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Zhi-Wei Sun

Publications and source records attributed to Zhi-Wei Sun.

At least 73 records · Page 4Linked to original sources

arXiv2023

Some new series for $1/π$ motivated by congruences

In this paper, we deduce a family of six new series for $1/π$; for example, $$\sum_{n=0}^\infty\frac{41673840n+4777111}{5780^n}W_n\left(\frac{1444}{1445}\right) =\frac{147758475}{\sqrt{95}\,π}$$ where $W_n(x)=\sum_{k=0}^n\binom nk\binom{n+k}k\binom{2k}k\binom{2(n-k)}{n-k}x^k$. To do so, we transform our series to series of the type $$\sum_{n=0}^\infty\frac{an+b}{m^n}\sum_{k=0}^n\binom nk^4$$ studied by Cooper in 2012. In addition, we pose $17$ new series for $1/π$ motivated by congruences; for example, we conjecture that $$\sum_{k=0}^\infty\frac{4290k+367}{3136^k}\binom{2k}kT_k(14,1)T_k(17,16)=\frac{5390}π,$$ where $T_k(b,c)$ is the coefficient of $x^k$ in the expansion of $(x^2+bx+c)^k$.

math.NT↗

arXiv2023

On determinants involving second-order recurrent sequences

Let $A$ and $B$ be complex numbers, and let $(w_n)_{n\ge0}$ be a sequence of complex numbers with $w_{n+1}=Aw_n-Bw_{n-1}$ for all $n=1,2,3,\ldots$. When $w_0=0$ and $w_1=1$, the sequence $(w_n)_{n\ge0}$ is just the Lucas sequence $(u_n(A,B))_{n\ge0}$. In this paper, we evaluate the determinants $$\det[w_{|j-k|}]_{1\le j,k\le n}\ \ \text{and}\ \ \det[w_{|j-k+1|}]_{1\le j,k\le n}.$$ In particular, we have $$\det[u_{|j-k|}(A,B)]_{1\le j,k\le n}=(-1)^{n-1}u_{n-1}(2A,(B+1)^2).$$ When $B=-1$ and $2\mid n$, we also determine the characteristic polynomial of the matrix $[w_{j+k}]_{0\le j,k\le n-1}$.

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arXiv2023

Evaluations of some Toeplitz-type determinants

In this paper we evaluate some Toeplitz-type determinants. Let $n>1$ be an integer. We prove the following two basic identities: \begin{align*} \det{[j-k+δ_{jk}]_{1\leq j,k\leq n}}&=1+\frac{n^2(n^2-1)}{12}, \\ \det{[|j-k|+δ_{jk}]_{1\leq j,k\leq n}}&= \begin{cases} \frac{1+(-1)^{(n-1)/2}n}{2}&\text{if}\ 2\nmid n,\\ \frac{1+(-1)^{n/2}}{2}&\text{if}\ 2\mid n, \end{cases} \end{align*} where $δ_{jk}$ is the Kronecker delta. For complex numbers $a,b,c$ with $b\not=0$ and $a^2\not=4b$, and the sequence $(w_m)_{m\in\mathbb Z}$ with $w_{k+1}=aw_k-bw_{k-1}$ for all $k\in\mathbb Z$, we establish the identity $$\det[w_{j-k}+cδ_{jk}]_{1\le j,k\le n} =c^n+c^{n-1}nw_0+c^{n-2}(w_1^2-aw_0w_1+bw_0^2)\frac{u_n^2b^{1-n}-n^2}{a^2-4b},$$ where $u_0=0$, $u_1=1$ and $u_{k+1}=au_k-bu_{k-1}$ for all $k=1,2,\ldots$.

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arXiv2022

On two new kinds of restricted sumsets

Let $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n\}$$ and $$C(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_i\in A_i\ (1\le i\le n),\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n,\ \text{and}\ a_n\not=a_1\}$$ recently introduced by the second author, when $G$ is the additive group of a field we obtain lower bounds for $|L(A_1,\ldots,A_n)|$ and $|C(A_1,\ldots,A_n)|$ via the polynomial method. Moreover, when $G$ is torsion-free and $A_1=\cdots=A_n$, we determine completely when $|L(A_1,\ldots,A_n)|$ or $|C(A_1,\ldots,A_n)|$ attains its lower bound.

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arXiv2022

A new theorem on quadratic residues modulo primes

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $b\in\mathbb Z$ and $\varepsilon\in\{\pm 1\}$. We mainly prove that $$\left|\left\{N_p(a,b):\ 1 \{ax^2+b\}_p$, and $\{m\}_p$ with $m\in\mathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.

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arXiv2022

A novel permanent identity with applications

Let $n$ be a positive integer, and define the rational function $S(x_1,\ldots,x_{2n})$ as the permanent of the matrix $[x_{j,k}]_{1\le j,k\le 2n}$, where $$x_{j,k}=\begin{cases}(x_j+x_k)/(x_j-x_k)&\text{if}\ j\not=k,\\1&\text{if}\ j=k.\end{cases}$$ We give an explicit formula for $S(x_1,\ldots,x_{2n})$ which has the following consequence: If one of the variables $x_1,\ldots,x_{2n}$ takes zero, then $S(x_1,\ldots,x_{2n})$ vanishes, i.e., $$\sum_{τ\in S_{2n}}\prod_{j=1\atop τ(j)\not=j}^{2n}\frac{x_j+x_{τ(j)}}{x_j-x_{τ(j)}}=0,$$ where we view an empty product $\prod_{i\in\emptyset}a_i$ as $1$. As an application, we show that if $ζ$ is a primitive $2n$-th root of unity then $$\sum_{τ\in S_{2n}}\prod_{j=1\atop τ(j)\not=j}^{2n}\frac{1+ζ^{j-τ(j)}}{1-ζ^{j-τ(j)}}=((2n-1)!!)^2$$ as conjectured by Z.-W. Sun.

math.CO↗

arXiv2022

Some parametric congruences involving generalized central trinomial coefficients

For $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the $n$th generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. In particular, $T_n=T_n(1,1)$ is the central trinomial coefficient. In this paper, we mainly establish some parametric congruences involving generalized central trinomial coefficients. As consequences, we prove that for any prime $p>3$ $$ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}}{12^k}T_k\equiv\left(\frac{p}{3}\right)\frac{3^{p-1}+3}{4}\pmod{p^2} $$ and $$ \sum_{k=0}^{p-1}\frac{T_kH_k}{3^k}\equiv\frac{3+\left(\frac{p}{3}\right)}{2}-p\left(1+\left(\frac{p}{3}\right)\right)\pmod{p^2}, $$ where $(-)$ denotes the Legendre symbol and $H_k:=\sum_{j=1}^k1/j$ denotes the $k$th harmonic number. These confirm two conjectural congruences of the second author.

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arXiv2022

Proof of a conjecture involving derangements and roots of unity

Let $n>1$ be an odd integer. For any primitive $n$-th root $ζ$ of unity in the complex field. Via the Engenvector-eigenvalue Identity, we show that $$\sum_{τ\in D(n-1)}\mathrm{sign}(τ)\prod_{j=1}^{n-1}\frac{1+ζ^{j-τ(j)}}{1-ζ^{j-τ(j)}} =(-1)^{\frac{n-1}{2}}\frac{((n-2)!!)^2}{n}, $$ where $D(n-1)$ is the set of all derangements of $1,\ldots,n-1$. This confirms a previous conjecture of Z.-W. Sun. Moreover, for each $δ=0,1$ we determine the value of $\det[x+m_{jk}]_{1\le j,k\le n}$ completely, where $$m_{jk}=\begin{cases}(1+ζ^{j-k})/(1-ζ^{j-k})&\text{if}\ j\not=k,\\δ&\text{if}\ j=k. \end{cases}$$

math.CO↗

arXiv2022

Arithmetic properties of some permanents

In this paper we study arithmetic properties of some permanents, many of which involve trigonometric functions. For any primitive $n$-th root $ζ$ of unity, we obtain closed formulas for the permanents $$\mathrm{per}\left[1-ζ^jx_k\right]_{1\le j,k\le n}\ \ \text{and}\ \ \mathrm{per}\left[\frac1{1-ζ^{j-k}x}\right]_{1\le j,k\le n}.$$ Another typical result states that for any odd integer $n>1$ we have $$t_n:=\frac1{\sqrt n}\mathrm{per}\left[\tanπ\frac{jk}n\right]_{1\le j,k\le (n-1)/2}\in\mathbb Z,$$ and that $t_p\equiv(-1)^{(p+1)/2}\pmod p$ for any odd prime $p$. We also pose several conjectures for further research; for example, we conjecture that $$\mathrm{per}[|j-k|]_{1\le j,k\le p}\equiv-\frac12\pmod p$$ for any odd prime $p$.

math.GM↗

arXiv2022

New type series for powers of $π$

Motivated by Ramanujan-type series and Zeilberger-type series, in this paper we investigate two new types of series for powers of $π$. For example, we prove that $$\sum_{k=0}^\infty(198k^2-425k+210)\frac{k^3\binom{2k}k^3}{4096^k}=-\frac1{21π}$$ and $$\sum_{k=0}^\infty\frac{198k^2-227k+47}{\binom{2k}k^3}=\frac{3264-4π^2}{63}.$$ We also pose many conjectures in this new direction.

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arXiv2022

On Motzkin numbers and central trinomial coefficients

The Motzkin numbers $M_n=\sum_{k=0}^n\binom n{2k}\binom{2k}k/(k+1)$ $(n=0,1,2,\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\ldots)$ given by the constant term of $(1+x+x^{-1})^n$, have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with $n$ any positive integer: $$\frac2n\sum_{k=1}^n(2k+1)M_k^2\in\mathbb Z,$$ $$\frac{n^2(n^2-1)}6\,\bigg|\,\sum_{k=0}^{n-1}k(k+1)(8k+9)T_kT_{k+1},$$ and also $$\sum_{k=0}^{n-1}(k+1)(k+2)(2k+3)M_k^23^{n-1-k}=n(n+1)(n+2)M_nM_{n-1}.$$

math.CO↗

arXiv2022

Sums of four rational squares with certain restrictions

In this paper we mainly study sums of four rational squares with certain restrictions. Let $\mathbb Q_{\ge0}$ be the set of nonnegative rational numbers. We establish the following four-square theorem for rational numbers: For any $a,b,c,d\in\mathbb Q_{\ge0}$, each $r\in\mathbb Q_{\ge0}$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Q_{\ge0}$ such that $ax+by+cz+dw$ is a rational square (or a rational cube). This paper also contains many conjectures; for example, for any positive integers $a$ and $b$ with $\gcd(a,b)=1$, we conjecture that each $r\in\mathbb Q_{\ge0}$ can be written as $aw^4+bx^4+y^2+z^2$ with $w,x,y,z\in\mathbb Q$.

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arXiv2021

Trigonometric identities and quadratic residues

In this paper we obtain some novel identities involving trigonometric functions. Let $n$ be any positive odd integer. We show that $$\sum_{r=0}^{n-1}\frac1{1+\sin2π\frac{x+r}n+\cos2π\frac{x+r}n} =\frac{(-1)^{(n-1)/2}n}{1+(-1)^{(n-1)/2}\sin 2πx+\cos 2πx}$$ for any complex number with $x+1/2,x+(-1)^{(n-1)/2}/4\not\in\mathbb Z$, and $$\sum_{j,k=0}^{n-1}\frac1{\sin 2π\frac{x+j}n+\sin2π\frac{y+k}n}=\frac{(-1)^{(n-1)/2}n^2}{\sin 2πx+\sin2πy}$$ for all complex numbers $x$ and $y$ with $x+y,x-y-1/2\not\in\mathbb Z$. We also determine the values of $\prod_{k=1}^{(p-1)/2}(1+\tanπ\frac{k^2}p)$ and $\prod_{k=1}^{(p-1)/2}(1+\cotπ\frac{k^2}p)$ for any odd prime $p$. In addition, we pose several conjectures on the values of $G_p(x)=\prod_{k=1}^{(p-1)/2}(x-e^{2πik^2/p})$ with $p$ an odd prime and $x$ a root of unity.

math.CA↗

arXiv2021

On exponential diophantine equations over $\mathbb Q$ with few unknowns

In this paper we obtain three undecidable results for exponential diophantine equations over the field $\mathbb Q$ of rational numbers. For example, we prove that there is no algorithm to decide the solvability of a general exponential diophantine equation $F(x_1,\ldots,x_{8})=0$ over $\mathbb Q$ with eight unknowns.

math.NT↗

arXiv2021

Proof of a conjectural supercongruence modulo $p^5$

In this paper we prove the supercongruence $$\sum_{n=0}^{(p-1)/2}\frac{6n+1}{256^n}\binom{2n}n^3\equiv p(-1)^{(p-1)/2}+(-1)^{(p-1)/2}\frac{7}{24}p^4B_{p-3}\pmod{p^5}$$ for any prime $p>3$, which was conjectured by Sun in 2019.

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arXiv2021

Proof of some conjectural hypergeometric supercongruences via curious identities

In this paper, we prove several supercongruences conjectured by Z.-W. Sun ten years ago via certain strange hypergeometric identities. For example, for any prime $p>3$, we show that $$\sum_{k=0}^{p-1}\frac{\binom{4k}{2k+1}\binom{2k}k}{48^k}\equiv0\pmod{p^2},$$ and $$ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}\binom{3k}{k}}{24^k}\equiv\begin{cases}\binom{(2p-2)/3}{(p-1)/3}\pmod{p^2}\ &\mbox{if}\ p\equiv1\pmod{3},\\ p/\binom{(2p+2)/3}{(p+1)/3}\pmod{p^2}\ &\mbox{if}\ p\equiv2\pmod{3}.\end{cases} $$ We also obtain some other results of such types.

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arXiv2021

On partitions of integers with restrictions involving squares

In this paper, we study partitions of positive integers with restrictions involving squares. We mainly establish the following two results (which were conjectured by Sun in 2013): (i) Each positive integer $n$ can be written as $n=x+y+z$ with $x,y,z$ positive integers such that $x^2+y^2+z^2$ is a square, unless $n$ has the form $n=2^{a}3^{b}$ or $2^{a}7$ with $a$ and $b$ nonnegative integers. (ii) Each integer $n>7$ with $n\not=11,14,17$ can be written as $n=x+y+2z$ with $x,y,z$ positive integers such that $x^2+y^2+2z^2$ is a square.

math.NT↗

arXiv2021

On permutations of $\{1,\ldots,n\}$ and related topics

In this paper we study combinatorial aspects of permutations of $\{1,\ldots,n\}$ and related topics. In particular, we prove that there is a unique permutation $π$ of $\{1,\ldots,n\}$ such that all the numbers $k+π(k)$ ($k=1,\ldots,n$) are powers of two. We also show that $n\mid\text{per}[i^{j-1}]_{1\le i,j\le n}$ for any integer $n>2$. We conjecture that if a group $G$ contains no element of order among $2,\ldots,n+1$ then any $A\subseteq G$ with $|A|=n$ can be written as $\{a_1,\ldots,a_n\}$ with $a_1,a_2^2,\ldots,a_n^n$ pairwise distinct. This conjecture is confirmed when $G$ is a torsion-free abelian group. We also prove that for any finite subset $A$ of a torsion-free abelian group $G$ with $|A|=n>3$, there is a numbering $a_1,\ldots,a_n$ of all the elements of $A$ such that all the $n$ sums $$a_1+a_2+a_3,\ a_2+a_3+a_4,\ \ldots,\ a_{n-2}+a_{n-1}+a_n,\ a_{n-1}+a_n+a_1,\ a_n+a_1+a_2$$ are pairwise distinct.

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