arXiv subjects

Zhi-Hong Sun

Publications and source records attributed to Zhi-Hong Sun.

At least 37 records · Page 2Linked to original sources

arXiv2020

Congruences involving binomial coefficients and Apéry-like numbers

For $n=0,1,2,\ldots$ let $W_n=\sum_{k=0}^{[n/3]}\binom{2k}k \binom{3k}k\binom n{3k}(-3)^{n-3k}$, where $[x]$ is the greatest integer not exceeding $x$. Then $\{W_n\}$ is an Apéry-like sequence. In this paper we deduce many congruences involving $\{W_n\}$, in particular we determine $\sum_{k=0}^{p-1}\binom{2k}k\frac{W_k}{m^k}\pmod p$ for $m=-640332,-5292,-972,-108,-44,-27,-12,8,54,243$ by using binary quadratic forms, where $p>3$ is a prime. We also prove several congruences for generalized Apéry-like numbers, and pose 29 challenging conjectures on congruences involving binomial coefficients and Apéry-like numbers.

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arXiv2020

Super congruences concerning binomial coefficients and Apéry-like numbers

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 $\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\frac p{k+b}\pmod {p^2}$. For $n=0,1,2,\ldots$ let $D_n$ and $b_n$ be Domb and Almkvist-Zudilin numbers, respectively. We also establish congruences for $$\sum_{n=0}^{p-1}\frac{D_n}{16^n},\quad \sum_{n=0}^{p-1}\frac{D_n}{4^n}, \quad \sum_{n=0}^{p-1}\frac{b_n}{(-3)^n},\quad \sum_{n=0}^{p-1}\frac{b_n}{(-27)^n}\pmod {p^2}$$ in terms of certain binary quadratic forms.

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arXiv2019

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

Let $\Bbb Z$ and $\Bbb Z^+$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb Z^+$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2+dw(w+1)/2$ $(x,y,z,w\in\Bbb Z)$. In this paper, by using Ramanujan's theta functions $φ(q)$ and $ψ(q)$ we present many formulas and conjectures on $t(a,b,c,d;n)$.

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arXiv2018

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

Let $\Bbb Z$ 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 $ax^2+by^2+cz^2$, and let $t(a,b,c;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2$ $(x,y,z\in\Bbb Z)$. In this paper, by using Ramanujan's theta functions $φ(q)$ and $ψ(q)$ we reveal the relation between $t(2,3,3;n)$ and $N(1,3,3;n+1)$, and the relation between $t(1,1,6;n)$ and $N(1,1,3;n+1)$. We also obtain formulas for $t(a,3a,4b;n),$ $t(a,7a,4b;n),t(3a,5a,4b;n)$ and $t(a,15a,4b;n)$ under certain congruence conditions, where $a$ and $b$ are positive odd integers. In addition, we pose many conjectures on $t(a,b,c;n)$ for some special values of $(a,b,c)$.

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arXiv2018

A kind of orthogonal polynomials and related identities II

For $n=0,1,2,\ldots$ let $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k}$. In this paper we illustrate the connection between $\{d_n^{(r)}(x)\}$ and Meixner polynomials. New formulas and recurrence relations for $d_n^{(r)}(x)$ are obtained, and a new proof of the formula for $d_n^{(r)}(x)^2$ is also given. In addition, for $r>-\frac 12$ and $n\ge 2$ we show that $d_n^{(r)}(x)>\frac{(2x+1)^n}{n!}>0$ for $x>-\frac 12$, and $(-1)^nd_n^{(r)}(x)>0$ for $x<-\frac 12$.

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arXiv2017

Ramanujan's theta functions and sums of triangular numbers

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a_1,a_2,\ldots,a_k,n\in\Bbb N$ let $N(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by $a_1x_1^2+a_2x_2^2+\cdots+a_kx_k^2$, and let $t(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by $a_1\frac{x_1(x_1-1)}2+a_2\frac{x_2(x_2-1)}2+\cdots+a_k\frac{x_k(x_k-1)}2$ $(x_1,\ldots,x_k\in\Bbb Z$). In this paper, by using Ramanujan's theta functions $φ(q)$ and $ψ(q)$ we reveal many relations between $t(a_1,a_2,\ldots,a_k;n)$ and $N(a_1,a_2,\ldots,a_k;8n+a_1+\cdots+a_k)$ for $k=3,4$.

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arXiv2017

A kind of orthogonal polynomials and related identities

In this paper we introduce the polynomials $\{d_n^{(r)}(x)\}$ and $\{D_n^{(r)}(x)\}$ given by $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k} \ (n\ge 0)$, $D_0^{(r)}(x)=1,\ D_1^{(r)}(x)=x$ and $D_{n+1}^{(r)}(x)=xD_n^{(r)}(x)-n(n+2r)D_{n-1}^{(r)}(x)\ (n\ge 1).$ We show that $\{D_n^{(r)}(x)\}$ are orthogonal polynomials for $r>-\frac 12$, and establish many identities for $\{d_n^{(r)}(x)\}$ and $\{D_n^{(r)}(x)\}$, especially obtain a formula for $d_n^{(r)}(x)^2$ and the linearization formulas for $d_m^{(r)}(x)d_n^{(r)}(x)$ and $D_m^{(r)}(x)D_n^{(r)}(x)$. As an application we extend recent work of Sun and Guo.

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arXiv2016

New super congruences involving Bernoulli and Euler polynomials

Let $p>3$ be a prime, and let $a$ be a rational p-adic integer with $a\not\equiv 0\pmod p$. In this paper we establish congruences for $$\sum_{k=1}^{(p-1)/2}\frac{\binom ak\binom{-1-a}k}k, \quad\sum_{k=0}^{(p-1)/2}k\binom ak\binom{-1-a}k \quad\text{and}\quad\sum_{k=0}^{(p-1)/2}\frac{\binom ak\binom{-1-a}k}{2k-1}\pmod {p^2}$$ in terms of Bernoulli and Euler polynomials. We also give some transformation formulas for congruences modulo $p^2$.

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arXiv2015

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

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\in\Bbb Z$). In this paper we reveal the connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\in\Bbb N$ and $2\nmid a$. We show that $$t(a,b,c,d;n)=\frac 23N(a,b,c,d;8n+a+b+c+d)-2N(a,b,c,d;2n+(a+b+c+d)/4)$$ for $(a,b,c,d)= (a,a,2a,8m),\ (a,3a,8k+2,8m+6),\ (a,3a,8m+4,8m+4)\ (n\equiv m+\frac{a-1}2 \pmod 2)$ and $(a,3a,16k+4,16m+4)\ (n\equiv \frac{a-1}2\pmod 2)$. We also obtain explicit formulas for $t(a,b,c,d;n)$ in the cases $(a,b,c,d)=(1,1,2,8),\ (1,1,2,16),(1,2,3,6),\ (1,3,4,12),\ (1,1,$ $3,4),\ (1,1,5,5),\ (1,5,5,5),\ (1,3,3,12),\ (1,1,1,12),\ (1,1,3,12)$ and $(1,3,3,4)$.

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arXiv2015

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

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arXiv2015

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

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\in\Bbb Z$). In this paper we obtain explicit formulas for $t(a,b,c,d;n)$ in the cases $(a,b,c,d)=(1,2,2,4),\ (1,2,4,4),\ (1,1,4,4),\ (1,4,4,4)$, $(1,3,9,9),\ (1,1,3,9)$, $(1,3,3,9)$, $(1,1,9,9),\ (1,9,9,9)$ and $(1,1,1,9).$

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arXiv2015

Congruences for Catalan-Larcombe-French numbers

Let $\{P_n\}$ be the Catalan-Larcombe-French numbers given by $P_0=1,\ P_1=8$ and $n^2P_n=8(3n^2-3n+1)P_{n-1}-128(n-1)^2P_{n-2}$ $(n\ge 2)$, and let $S_n=P_n/2^n$. In this paper we deduce congruences for $S_{mp^r}\pmod{p^{r+2}}$, $S_{mp^r-1}\pmod{p^r}$ and $S_{mp^r+1}\pmod{p^{2r}}$, where $p$ is an odd prime and $m,r$ are positive integers. We also prove that $S_{(p^2-1)/2}\equiv 0\pmod {p^2}$ for any prime $p\equiv 5,7\pmod 8$, and show that $\{S_m\}$ is log-convex.

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arXiv2015

Turán's problem and Ramsey numbers for trees

Let $T_n^1=(V,E_1)$ and $T_n^2=(V,E_2)$ be the trees on $n$ vertices with $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_1=\{v_0v_1,\ldots,v_0v_{n-3},v_{n-4}v_{n-2},v_{n-3}v_{n-1}\}$, and $E_2=\{v_0v_1,\ldots,$ $v_0v_{n-3},v_{n-3}v_{n-2}, v_{n-3}v_{n-1}\}$. In this paper, for $p\ge n\ge 5$ we obtain explicit formulas for $\ex(p;T_n^1)$ and $\ex(p;T_n^2)$, where $\ex(p;L)$ denotes the maximal number of edges in a graph of order $p$ not containing $L$ as a subgraph. Let $r(G\sb 1, G\sb 2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. In this paper we also obtain some explicit formulas for $r(T_m,T_n^i)$, where $i\in\{1,2\}$ and $T_m$ is a tree on $m$ vertices with $Δ(T_m)\le m-3$.

math.CO↗

arXiv2015

Congruences involving Franel and Catalan-Larcombe-French numbers

Let $\{f_n\}$ be the Franel numbers given by $f_n=\sum_{k=0}^n\binom nk^3$, and let $p>5$ be a prime. In this paper we mainly determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{f_k}{m^k}\pmod p$ for $m=5,-16,16,32,-49,50,96$. Let $S_n=\sum_{k=0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k}$. We also determine $\sum_{k=0}^{p-1} \binom{2k}k\frac{S_k}{m^k}\pmod p$ for $m=7,16,25,32,64,160,800,1600, 156832$.

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arXiv2015

Note on super congruences modulo $p^2$

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

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arXiv2015

Congruences for Domb and Almkvist-Zudilin numbers

In this paper we prove some transformation formulae for congruences modulo a prime and deduce some congruences for Domb numbers and Almkvist-Zudilin numbers. We also pose some conjectures on congruences modulo prime powers.

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arXiv2014

Turán's problem and generalized Ramsey numbers

Let $n,r,k,s$ be positive integers with $n,k\ge 2$. The generalized Ramsey number $R(n,r;k,s)$ is the smallest positive integer $p$ such that for every graph $G$ of order $p$, either $G$ contains a subgraph induced by $n$ vertices with at most $r-1$ edges, or the complement $\bar G$ of $G$ contains a subgraph induced by $k$ vertices with at most $s-1$ edges. In this paper we completely determine $R(n,n(n-1)/2-r;k,1)$ for $n\ge 4$ and $r\le n-2$, and pose several conjectures on Ramsey numbers.

math.CO↗

arXiv2014

Ramsey numbers for trees

For $n\ge 5$ let $T_n'$ denote the unique tree on $n$ vertices with $Δ(T_n')=n-2$, and let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\{v_0,v_1,\ldots,$ $v_{n-1}\}$ and $E=\{v_0v_1,\ldots,v_0v_{n-3},$ $v_{n-3}v_{n-2},v_{n-2}v_{n-1}\}$. In this paper we evaluate the Ramsey numbers $r(G_m,T_n')$ and $r(G_m,T_n^*)$, where $G_m$ is a connected graph of order $m$. As examples, for $n\ge 8$ we have $r(T_n',T_n^*)=r(T_n^*,T_n^*)=2n-5$, for $n>m\ge 7$ we have $r(K_{1,m-1},T_n^*)=m+n-3$ or $m+n-4$ according as $m-1\mid (n-3)$ or $m-1\nmid (n-3)$, for $m\ge 7$ and $n\ge (m-3)^2+2$ we have $r(T_m^*,T_n^*)=m+n-3$ or $m+n-4$ according as $m-1\mid (n-3)$ or $m-1\nmid (n-3)$.

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