Zhi-Wei Sun

Multiple Clausen values and deformed Apéry-like series

Zhi-Wei Sun, Yajun Zhou

With generalized central binomial coefficients $(x 2 x) := \frac{Γ ( 2 x + 1 )}{[ Γ ( x + 1 ) ] ^{2}}$ defined through Euler's gamma function, we represent deformed Apéry-like series \[ \math…

Some new results on determinants and permanents

Bo Jiang, Zhi-Wei Sun

In this paper we confirm several conjectures on determinants and permanents. For example, we prove that for any prime $p \equiv 3 (mod 4)$ the number $2\det[a_{jk}]_{0\le j,k\le (p-1)…

Various conjectural series identities

Zhi-Wei Sun

In this paper we collect over 150 new series identities (involving binomial coefficients) conjectured by the author in 2026. The values involved are related to $π$ or Riemann's zet…

Series involving central binomial coefficients and higher-order harmonic numbers

Zhi-Wei Sun, Yajun Zhou

We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence…

On series identities involving $(k 4 k)$ and harmonic numbers

Bo Jiang, Zhi-Wei Sun

The harmonic numbers are those $H_{n} = \sum_{0 < k \leq n} \frac{1}{k} (n = 0, 1, 2, \dots)$ . In this paper we confirm over ten conjectural series identities with summands involving the binomial…

Evaluation of a determinant involving Legendre symbols

Chen-Kai Ren, Zhi-Wei Sun

Let $p > 3$ be a prime, and let $(\frac{\cdot}{p})$ be the Legendre symbol. Let $A_{p} (x)$ denote the matrix $[x + a_{ij}]_{1 ⩽ i, j ⩽ (p - 1) /2}$ , where $$ a_{ij}=\begin{cases}…

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

Zhi-Wei Sun, Yajun Zhou

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…

New series involving binomial coefficients (III)

Zhi-Wei Sun

We evaluate some series with summands involving a single binomial coefficient $(3 k 6 k)$ . For example, we prove that $$\sum_{k=0}^\infty\frac{(63k^2+78k+22)8^k}{(2k+1)(6k+1)(…

A family of polynomials and related congruences and series

Zhi-Wei Sun

In this paper we study a family of polynomials $S_{n}^{(m)} (x) := i, j = 0 \sum n (i n)^{m} (j n)^{m} (i i + j) x^{i + j} (m, n = 0, 1, 2, \dots) .$ For example, we show that $$\sum_{k=…

On $p$ -adic congruences involving $d$

Bo Jiang, Zhi-Wei Sun

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…

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

Deyi Chen, Zhi-Wei Sun

Let $p = 2 n + 1$ 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 $…

On determinants involving $(\frac{j ^{2} - k ^{2}}{p})$ and $(\frac{j k}{p})$

Deyi Chen, Zhi-Wei Sun

Let $p$ be an odd prime and let $(\frac{\cdot}{p})$ be the Legendre symbol. In this paper, we study the determinant $$\det\left[\left(\frac{j^2-k^2}p\right)+\left(\frac{jk}p\right)w\…

On a determinant involving linear combinations of Legendre symbols

Keqin Liu, Zhi-Wei Sun, Li-Yuan Wang

In this paper, we prove a conjecture of the second author by evaluating the determinant $$\det\left[x + \left(\frac{i-j}p\right) + \left(\frac ip\right)y + \left(\frac jp\right)z +…

Some determinants involving binary forms

Yue-Feng She, Zhi-Wei Sun

In this paper, we study arithmetic properties of certain determinants involving powers of $i^{2} + c ij + d j^{2}$ , where $c$ and $d$ are integers. For example, for any odd integer $n > 1$ wit…

math.NT2024★ 1 cited

Problems and results on determinants involving Legendre symbols

Zhi-Wei Sun

In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. We deduce some new results in this direction; for example, we prove that for an…