arXiv subjects

Dian-Wang Hu

Publications and source records attributed to Dian-Wang Hu.

2 recordsLinked to original sources

arXiv2026

A Congruence Conjecture of Z.-W. Sun for Reciprocal Central Binomial Sums

Let $p>5$ be a prime. We prove a conjecture of Z.-W. Sun asserting that $ \sum_{k=1}^{p-1}\frac{1}{k^4\binom{2k}{k}} \equiv \frac{H_{p-1}}{p^3}-\frac{7}{45}pB_{p-5}\pmod{p^2}.$ As a consequence, we obtain $ \sum_{k=1}^{p-1} \frac{H_{k-1}^{(2)}}{k^2\binom{2k}{k}} \equiv \frac{H_{p-1}}{3p^3} +\frac{26}{135}pB_{p-5} \pmod{p^2},$ an equivalent conjecture of K. Hessami Pilehrood and T. Hessami Pilehrood. Here $H_n^{(m)}$ denotes the generalized harmonic numbers of order $m$, with $H_n=H_n^{(1)}$, and $B_n$ denotes the $n$th Bernoulli number. The proof combines $p$-adic congruences involving harmonic numbers, finite-sum identities, and complex residue calculations via the residue theorem.

math.NT↗

arXiv2020

Proof of some supercongruences concerning truncated hypergeometric series

In this paper, we prove some supercongruences concerning truncated hypergeometric series. For example, we show that for any prime $p>3$ and positive integer $r$, $$ \sum_{k=0}^{p^r-1}(3k+1)\frac{(\frac12)_k^3}{(1)_k^3}4^k\equiv p^r+\frac76p^{r+3}B_{p-3}\pmod{p^{r+4}} $$ and $$ \sum_{k=0}^{(p^r-1)/2}(4k+1)\frac{(\frac12)_k^4}{(1)_k^4}\equiv p^r+\frac76p^{r+3}B_{p-3}\pmod{p^{r+4}}, $$ where $(x)_k=x(x+1)\cdots(x+k-1)$ is the Pochhammer symbol and $B_0,B_1,B_2,\ldots$ are Bernoulli numbers. These two congruences confirm conjectures of Sun [Sci. China Math. 54 (2011), 2509--2535] and Guo [Adv. Appl. Math. 120 (2020), Art. 102078], respectively.

math.NT↗