arXiv subjects
Publications and source records attributed to Zhi-Hong Sun.
In this paper, we establish a general identity for three-term recurrence sequences and then give applications to orthogonal polynomials and Apéry-like numbers.
We establish weighted bilinear summation identities for two families of Apéry-like polynomials $g_n(x)$ and $v_n(x)$. The identities express weighted sums in terms of consecutive endpoint values and, when necessary, lower moments. For $g_n(x)^2$ we obtain identities with weights $(2n+1)^r$ for $1\le r\le4$; for $v_n(x)^2$ we treat the cubic and quintic weights. Combining these formulas with congruences for the endpoint values gives supercongruences modulo $p^3$ and $p^4$, together with special evaluations modulo $p^5$ and $p^7$, where $p$ is a prime greater than $3$. In particular, $$\sum_{n=0}^{p-1}(2n+1)^3v_n\!\left(\frac52\right)^2 \equiv 6p^4-\frac{143}{3}p^6\pmod {p^7},$$ confirming a congruence conjectured by Sun. The proofs use explicit quadratic telescoping identities and $p$-adic endpoint expansions.
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 congruences for $g_n(x)$ and $v_n(x)$ given by \begin{align*} &g_0(x)=1,\ g_1(x)=\frac{x+1}2,\ (n+1)^2g_{n+1}(x)=\Big(2n(n+1)+\frac{x+1}2\Big)g_n(x)-n^2g_{n-1}(x)\ (n\ge 1), \\&v_0(x)=1,\ v_1(x)=x,\ (n+1)^3v_{n+1}(x)=(2n+1)(n(n+1)+x)v_n(x)-n^3v_{n-1}(x)\ (n\ge 1).\end{align*}
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.
Let $p>3$ be a prime, $a_1,a_2,a_3\in\Bbb Z$ and let $N_p(x^3+a_1x^2+a_2x+a_3)$ denote the number of solutions to the congruence $x^3+a_1x^2+a_2x+a_3\equiv 0\pmod p$. In this paper, we give an explicit criterion for $N_p(x^3+a_1x^2+a_2x+a_3)=3$ via binary quadratic forms.
Let $\{A'_n\}$ be the Apéry numbers given by $A'_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k.$ For any prime $p\equiv 3\pmod 4$ we show that $A'_{\frac{p-1}2}\equiv \frac{p^2}3\binom{\frac{p-3}2}{\frac{p-3}4}^{-2}\pmod {p^3}$. Let $\{t_n\}$ be given by $$t_0=1,\ t_1=5\quad\hbox{and}\quad t_{n+1}=(8n^2+12n+5)t_n-4n^2(2n+1)^2t_{n-1}\ (n\ge 1).$$ We also obtain the congruences for $t_p\pmod {p^3},\ t_{p-1}\pmod {p^2}$ and $t_{\frac{p-1}2}\pmod {p^2}$, where $p$ is an odd prime.
Recently, using modular forms F. Beukers posed a unified method that can deal with a large number of supercongruences involving binomial coefficients and Apéry-like numbers. In this paper, we use Beukers' method to prove some conjectures of the first author concerning the congruences for $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k^3}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k\binom{6k}{3k}}{m^k}, \ \sum_{n=0}^{p-1}\frac{V_n}{m^n},\ \sum_{n=0}^{p-1}\frac{T_n}{m^n},\ \sum_{n=0}^{p-1}\frac{D_n}{m^n} $$ and $\sum_{n=0}^{p-1}(-1)^nA_n$ modulo $p^3$, where $p$ is an odd prime representable by some suitable binary quadratic form, $m$ is an integer not divisible by $p$, $V_n=\sum_{k=0}^n\binom{2k}k^2\binom{2n-2k}{n-k}^2$, $T_n=\sum_{k=0}^n\binom nk^2\binom{2k}n^2$, $D_n=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2n-2k}{n-k}$ and $A_n$ is the Apéry number given by $A_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2$.
For any prime $p>3$ and rational $p$-integers $b,c$ with $c(b^3-27c)\not\equiv 0\pmod p$ let $V_p(x^2+bx+\frac cx)$ be the residue-counts of $x^2+bx+\frac cx$ modulo $p$ as $x$ runs over $1,2,\ldots,p-1$. In this paper, we reveal the connection between $V_p(x^2+bx+\frac cx)$ and the number of points on certain elliptic curve over the field $\Bbb F_p$.
For a prime $p>3$ and $a\in \Bbb Z$ with $p\nmid a$ let $V_p(x^2+\frac ax)$ be the residue-counts of $x^2+\frac ax$ modulo $p$ as $x$ runs over $1,2,\ldots,p-1$. In this paper, we obtain an explicit formula for $V_p(x^2+\frac ax)$, which is concerned with cubic residues and binary quadratic forms.
Let $r(G_1, G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V(S(n_1,n_2))=\{v_0,v_1,\ldots,v_{n_1},w_0,w_1,\ldots,w_{n_2}\}$ and $E(S(n_1,n_2))=\{v_0v_1,\ldots,v_0v_{n_1},v_0w_0,w_0w_1,\ldots,w_0w_{n_2}\}$. In this paper we determine $r(K_{1,m-1},$ $S(n_1,n_2))$ under certain conditions. For $n\ge 6$ let $T_n^3=S(n-5,3)$, $T_n^{''}=(V,E_2)$ and $T_n^{'''} =(V,E_3)$, where $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_2=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2},$ $v_2v_{n-1}\}$ and $E_3=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_2v_{n-2},v_3v_{n-1}\}$. We also obtain explicit formulas for $r$ $(K_{1,m-1},T_n)$, $r(T_m',T_n)$ $(n\ge m+3)$, $r(T_n,T_n)$, $r(T_n',T_n)$ and $r(P_n,T_n)$, where $T_n\in\{T_n'',T_n''',T_n^3\}$, $P_n$ is the path on $n$ vertices and $T_n'$ is the unique tree with $n$ vertices and maximal degree $n-2$.
Let $p>3$ be a prime. In this paper, we obtain the congruences for $$\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^3}{(-8)^k},\ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{3k}k}{(-192)^k},\ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{4k}{2k}}{(-144)^k}\ \text{and} \ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{4k}{2k}}{648^k}$$ modulo $p^2$, and partial results for $\sum_{k=0}^{(p-1)/2} \binom{2k}k^3\frac{w(k)}{m^k}$ modulo $p^2$, where $m\in\{1,16,-64,256,-512,4096\}$ and $w(k)\in\{k^2,k^3,\frac 1{k+1},\frac 1{(k+1)^2},\frac 1{(k+1)^3}, \frac 1{2k-1},\frac 1{k+2}\}$.
If $f(x,y)$ is a real function satisfying $y>0$ and $\sum_{r=0}^{n-1}f(x+ry,ny)=f(x,y)$ for $n=1,2,3,\ldots$, we say that $f(x,y)$ is an invariant function. Many special functions including Bernoulli polynomials, Gamma function and Hurwitz zeta function are related to invariant functions. In this paper we systematically investigate the properties of invariant functions.
Let $p$ be an odd prime, and let $a$ be a rational $p$-adic integer with $a\not\equiv 0\pmod p$. In this paper, using WZ method we establish the congruences for $\sum_{k=0}^{p-1} \binom ak^2(-1)^k(1-\frac 2ak)$ modulo $p^2$ and $\sum_{k=0}^{p-1} \binom ak^r(1-\frac 2ak)^s$ modulo $p^4$, where $r\in\{3,4\}$ and $s\in\{1,3\}$.
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 $$\sum_{k=0}^{p-1} \binom ak\binom{-1-a}k\binom{2k}k\frac {w(k)}{4^k},$$ where $$w(k)=1,\frac 1{k+1},\frac 1{(k+1)^2},\frac 1{(k+1)^3},\frac 1{2k-1},\frac 1{k+2}, \frac 1{k+3}, k,k^2,k^3,\frac 1{a+k},\frac 1{a+k-1}.$$ As consequences, taking $a=-\frac 12,-\frac 13,-\frac 14,-\frac 16$ we deduce many congruences modulo $p^3$ and so solve some conjectures posed by the author earlier.
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.
In this paper, we pose many challenging conjectures on congruences involving binomial coefficients and Apéry-like numbers.
In this paper we present many congruences for several Apéry-like sequences.
In this paper we present many results and conjectures on congruences involving two types of Apéry-like sequences $\{G_n(x)\}$ and $\{V_n(x)\}$.