Zhi-Hong Sun

activity

20102026

most citedCongruences concerning Legendre polynomials

47 citations · 109 across the 54 of their papers we have counts for

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45 papers · 1 filter

math.NT2026

Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers

Zhi-Hong Sun

In this paper, we first extend the Christoffel-Darboux formula for orthogonal polynomials to general three-term recurrence sequences, and then investigate the identities and congru…

math.NT2026

Curious identities involving Legendre polynomials and Apéry-like numbers

Li-Li Cui, Zhi-Hong Sun

In this paper, we establish some curious identities involving Legendre polynomials and the first kind of Apéry-like numbers. As applications, many new supercongruences are deduced.

math.NT2022

Congruences concerning binomial coefficients and binary quadratic forms

Zhi-Hong Sun

Let $p > 3$ be a prime. In this paper, we obtain the congruences for $$\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^3}{(-8)^k},\ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{3k}k}{(-192)^k…

math.NT2022

Supercongruences for sums involving $(k a)^{m}$

Zhi-Hong Sun

Let $p$ be an odd prime, and let $a$ be a rational $p$ -adic integer with $a \neq \equiv 0 (mod p)$ . In this paper, using WZ method we establish the congruences for $\sum_{k=0}^{p-1} \…

math.NT2022

Congruences for sums involving products of three binomial coefficients

Zhi-Hong Sun

Let $p > 3$ be a prime, and let $a$ be a rational $p$ -adic integer, using WZ method we establish the congruences modulo $p^{3}$ for $$\sum_{k=0}^{p-1} \binom ak\binom{-1-a}k\binom{2k}k…

math.NT2021

New conjectures involving binomial coefficients and Apéry-like numbers

Zhi-Hong Sun

In this paper, we pose lots of challenging conjectures on congruences for the sums involving binomial coefficients and Apéry-like numbers modulo $p^{3}$ , where $p$ is an odd prime.

math.NT2020

Conjectures on congruences involving binomial coefficients and Apéry-like numbers

Zhi-Hong Sun

In this paper, we pose many challenging conjectures on congruences involving binomial coefficients and Apéry-like numbers.

math.NT2020★ 2 cited

Congruences for two types of Apery-like sequences

Zhi-Hong Sun

In this paper we present many results and conjectures on congruences involving two types of Apéry-like sequences ${G_{n} (x)}$ and ${V_{n} (x)}$ .

math.NT2020★ 5 cited

New congruences involving Apéry-like numbers

Zhi-Hong Sun

In this paper we present many congruences for several Apéry-like sequences.

math.NT2020★ 2 cited

Super congruences concerning binomial coefficients and Apéry-like numbers

Zhi-Hong Sun

Let $p$ be a prime with $p > 3$ , and let $a, b$ be two rational $p -$ integers. In this paper we present general congruences for $\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\frac p{k+b}\pmod…

math.NT2018

Ramanujan's theta functions and linear combinations of three triangular numbers

Zhi-Hong Sun

Let $Z$ be the set of integers. For positive integers $a, b, c$ and $n$ let $N (a, b, c; n)$ be the number of representations of $n$ by $a x^{2} + b y^{2} + c z^{2}$ , and let $t (a, b, c; n)$ be the…

math.NT2018★ 1 cited

Ramanujan's theta functions and linear combinations of four triangular numbers

Zhi-Hong Sun

Let $Z$ and $Z^{+}$ be the set of integers and the set of positive integers, respectively. For $a, b, c, d, n \in Z^{+}$ let $t (a, b, c, d; n)$ be the number of representations of…

math.NT2018

Congruences involving binomial coefficients and Apéry-like numbers

Zhi-Hong Sun

For $n = 0, 1, 2, \dots$ let $W_{n} = \sum_{k = 0}^{[n /3]} (k 2 k) (k 3 k) (3 k n) (- 3)^{n - 3 k}$ , where $[x]$ is the greatest integer not exceeding $x$ . Then ${W_{n}}$ is an Apéry-l…

math.NT2016

A kind of orthogonal polynomials and related identities

Zhi-Hong Sun

In this paper we introduce the polynomials ${d_{n}^{(r)} (x)}$ and ${D_{n}^{(r)} (x)}$ given by $d_{n}^{(r)} (x) = \sum_{k = 0}^{n} (k x + r + k) (n - k x - r) (n \geq 0)$ , $D_0^{(r)}(x)=1,…

math.NT2016

New super congruences involving Bernoulli and Euler polynomials

Zhi-Hong Sun

Let $p > 3$ be a prime, and let $a$ be a rational p-adic integer with $a \neq \equiv 0 (mod p)$ . In this paper we establish congruences for $$\sum_{k=1}^{(p-1)/2}\frac{\binom ak\binom{-…

math.NT2016★ 1 cited

Ramanujan's theta functions and sums of triangular numbers

Zhi-Hong Sun

Let $Z$ and $N$ be the set of integers and the set of positive integers, respectively. For $a_{1}, a_{2}, \dots, a_{k}, n \in N$ let $N (a_{1}, a_{2}, \dots, a_{k}; n)$ be the number of…

math.NT2015

On the number of representations of n as a linear combination of four triangular numbers II

Min Wang, Zhi-Hong Sun

Let $Z$ and $N$ be the set of integers and the set of positive integers, respectively. For $a, b, c, d, n \in N$ let $N (a, b, c, d; n)$ be the number of representations of $n$ …

math.NT2015★ 2 cited

Some relations between t(a,b,c,d;n) and N(a,b,c,d;n)

Zhi-Hong Sun

Let $Z$ and $N$ be the set of integers and the set of positive integers, respectively. For $a, b, c, d, n \in N$ let $N (a, b, c, d; n)$ be the number of representations of $n$ …

math.NT2015★ 4 cited

On the number of representations of n as a linear combination of four triangular numbers

Min Wang, Zhi-Hong Sun

Let $Z$ and $N$ be the set of integers and the set of positive integers, respectively. For $a, b, c, d, n \in N$ let $t (a, b, c, d; n)$ be the number of representations of $n$ …

math.NT2015★ 1 cited

Congruences for Catalan-Larcombe-French numbers

Xiao-Juan Ji, Zhi-Hong Sun

Let ${P_{n}}$ be the Catalan-Larcombe-French numbers given by $P_{0} = 1, P_{1} = 8$ and $n^{2} P_{n} = 8 (3 n^{2} - 3 n + 1) P_{n - 1} - 128 (n - 1)^{2} P_{n - 2}$ $(n \geq 2)$ , and let $S_{n} = P_{n} / 2^{n}$ . In this paper we…

math.NT2015★ 1 cited

Note on super congruences modulo $p^{2}$

Zhi-Hong Sun

Let $p$ be an odd prime, and let $m$ be an integer with $p ∤ m$ . In this paper show that $$\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom ak\binom{-1-a}k}{m^k} \equiv 0\pmod p \quad\hb…

math.NT2015★ 3 cited

Congruences involving Franel and Catalan-Larcombe-French numbers

Zhi-Hong Sun

Let ${f_{n}}$ be the Franel numbers given by $f_{n} = \sum_{k = 0}^{n} (k n)^{3}$ , and let $p > 5$ be a prime. In this paper we mainly determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{f_k}{m^k}…

math.NT2014★ 9 cited

Super congruences involving Bernoulli and Euler polynomials

Zhi-Hong Sun

Let $p > 3$ be a prime, and let $a$ be a rational p-adic integer. Let ${B_{n} (x)}$ and ${E_{n} (x)}$ denote the Bernoulli polynomials and Euler polynomials, respectively. In this pape…

math.NT2014★ 2 cited

Recurrence formulas with gaps for Bernoulli and Euler polynomials

Zhi-Hong Sun

Let ${B_{n}}$ , ${B_{n} (x)}$ and ${E_{n} (x)}$ be the Bernoulli numbers, Bernoulli polynomials and Euler polynomials, respectively. In this paper we mainly establish formulas for $\s…

Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers

Curious identities involving Legendre polynomials and Apéry-like numbers

Congruences concerning binomial coefficients and binary quadratic forms

Supercongruences for sums involving (ka​)m

Congruences for sums involving products of three binomial coefficients

New conjectures involving binomial coefficients and Apéry-like numbers

Conjectures on congruences involving binomial coefficients and Apéry-like numbers

Congruences for two types of Apery-like sequences

New congruences involving Apéry-like numbers

Super congruences concerning binomial coefficients and Apéry-like numbers

Ramanujan's theta functions and linear combinations of three triangular numbers

Ramanujan's theta functions and linear combinations of four triangular numbers

Congruences involving binomial coefficients and Apéry-like numbers

A kind of orthogonal polynomials and related identities

New super congruences involving Bernoulli and Euler polynomials

Ramanujan's theta functions and sums of triangular numbers

On the number of representations of n as a linear combination of four triangular numbers II

Some relations between t(a,b,c,d;n) and N(a,b,c,d;n)

On the number of representations of n as a linear combination of four triangular numbers

Congruences for Catalan-Larcombe-French numbers

Note on super congruences modulo p2

Congruences involving Franel and Catalan-Larcombe-French numbers

Super congruences involving Bernoulli and Euler polynomials

Recurrence formulas with gaps for Bernoulli and Euler polynomials

Supercongruences for sums involving $(k a)^{m}$

Note on super congruences modulo $p^{2}$