arXiv subjects

Victor J. W. Guo

Publications and source records attributed to Victor J. W. Guo.

At least 37 records · Page 2Linked to original sources

arXiv2019

$q$-Analogues of Dwork-type supercongruences

In 1997, Van Hamme conjectured 13 Ramanujan-type supercongruences. All of the 13 supercongruences have been confirmed by using a wide range of methods. In 2015, Swisher conjectured Dwork-type supercongruences related to the first 12 supercongruences of Van Hamme. Here we prove that the (C.3) and (J.3) supercongruences of Swisher are true modulo $p^{3r}$ (the original modulus is $p^{4r}$) by establishing $q$-analogues of them. Our proof will use the creative microscoping method, recently introduced by the author in collaboration with Zudilin. We also raise conjectures on $q$-analogues of an equivalent form of the (M.2) supercongruence of Van Hamme, partially answering a question at the end of [Adv. Math. 346 (2019), 329--358].

math.NT↗

arXiv2019

A family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial

We prove a two-parameter family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Crucial ingredients in our proof are George Andrews' multiseries extension of the Watson transformation, and a Karlsson--Minton type summation for very-well-poised basic hypergeometric series due to George Gasper. The new family of $q$-congruences is then used to prove two conjectures posed earlier by the authors.

math.NT↗

arXiv2019

Proof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial

By means of the $q$-Zeilberger algorithm, we prove a basic hypergeometric supercongruence modulo the fifth power of the cyclotomic polynomial $Φ_n(q)$. This result appears to be quite unique, as in the existing literature so far no basic hypergeometric supercongruences modulo a power greater than the fourth of a cyclotomic polynomial have been proved. We also establish a couple of related results, including a parametric supercongruence.

math.NT↗

arXiv2019

On a $q$-deformation of modular forms

There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work arXiv:1803.01830, such partial sums are related to the radial asymptotics of infinite $q$-hypergeometric sums at roots of unity. Here we combine the two features to construct a hypergeometric $q$-deformation of two CM modular forms of weight 3 and discuss the corresponding $q$-congruences.

math.NT↗

arXiv2019

Some congruences related to a congruence of Van Hamme

We establish some supercongruences related to a supercongruence of Van Hamme, such as \begin{align*} \sum_{k=0}^{(p+1)/2} (-1)^k (4k-1)\frac{(-\frac{1}{2})_k^3}{k!^3} &\equiv p(-1)^{(p+1)/2}+p^3(2-E_{p-3})\pmod{p^{4}},\\ \sum_{k=0}^{(p+1)/2} (4k-1)^5 \frac{(-\frac{1}{2})_k^4}{k!^4} &\equiv 16p\pmod{p^{4}}, \end{align*} where $p$ is an odd prime and $E_{p-3}$ is the $(p-3)$-th Euler number. Our proof uses some congruences of Z.-W. Sun, the Wilf--Zeilberger method, Whipple's $_7F_6$ transformation, and the software package {\tt Sigma} developed by Schneider. We also put forward two related conjectures.

math.CO↗

arXiv2019

Some new $q$-congruences for truncated basic hypergeometric series

We provide several new $q$-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews' multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the fourth power of a cyclotomic polynomial.

math.NT↗

arXiv2019

A $q$-microscope for supercongruences

By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a $q$-analogue of Ramanujan's formula $$ \sum_{n=0}^\infty\frac{\binom{4n}{2n}{\binom{2n}{n}}^2}{2^{8n}3^{2n}}\,(8n+1) =\frac{2\sqrt{3}}π, $$ of the two supercongruences $$ S(p-1)\equiv p\biggl(\frac{-3}p\biggr)\pmod{p^3} \quad\text{and}\quad S\Bigl(\frac{p-1}2\Bigr) \equiv p\biggl(\frac{-3}p\biggr)\pmod{p^3}, $$ valid for all primes $p>3$, where $S(N)$ denotes the truncation of the infinite sum at the $N$-th place and $(\frac{-3}{\cdot})$ stands for the quadratic character modulo $3$.

math.NT↗

arXiv2019

Factors of some truncated basic hypergeometric series

We prove that certain basic hypergeometric series truncated at $k=n-1$ have the factor $Φ_n(q)^2$, where $Φ_n(q)$ is the $n$-th cyclotomic polynomial. This confirms two recent conjectures of the author and Zudilin. We also put forward some conjectures on $q$-congruences modulo $Φ_n(q)^2$.

math.NT↗

arXiv2018

Some $q$-congruences with parameters

Let $Φ_n(q)$ be the $n$-th cyclotomic polynomial in $q$. Recently, the author and Zudilin provide a creative microscoping method to prove some $q$-supercongruences mainly modulo $Φ_n(q)^3$ by introducing an additional parameter $a$. In this paper, we use this creative microscoping method to confirm some conjectures on $q$-supercongruences modulo $Φ_n(q)^2$. We also give some parameter-generalizations of known $q$-supercongruences. For instance, we present further generalizations of a $q$-analogue of a famous supercongruence of Rodriguez-Villegas: $$ \sum_{k=0}^{p-1}\frac{{2k\choose k}^2}{16^k} \equiv (-1)^{(p-1)/2}\pmod{p^2}\quad\text{for any odd prime $p$.} $$

math.NT↗

arXiv2018

Proofs of two conjectures on Catalan triangle numbers

We prove two conjectures on sums of products of Catalan triangle numbers, which were originally conjectured by Miana, Ohtsuka, and Romero [Discrete Math. 340 (2017), 2388--2397]. The first one is proved by using Zeilberger's algorithm, and the second one is proved by establishing its $q$-analogue.

math.CO↗

arXiv2018

Ramanujan-type formulae for $1/π$: $q$-analogues

The hypergeometric formulae designed by Ramanujan more than a century ago for efficient approximation of $π$, Archimedes' constant, remain an attractive object of arithmetic study. In this note we discuss some $q$-analogues of Ramanujan-type evaluations and of related supercongruences.

math.NT↗

arXiv2018

$q$-Analogues of two Ramanujan-type formulas for $1/π$

We give $q$-analogues of the following two Ramanujan-type formulas for $1/π$: \begin{align*} \sum_{k=0}^\infty (6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 4^k} =\frac{4}π \quad\text{and}\quad \sum_{k=0}^\infty (-1)^k(6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 8^k } =\frac{2\sqrt{2}}π. \end{align*} Our proof is based on two $q$-WZ pairs found by the first author in his earlier work.

math.NT↗

arXiv2018

q-Analogues of two "divergent" Ramanujan-type supercongruences

Guillera and Zudilin proved three "divergent" Ramanujan-type supercongruences by means of the Wilf-Zeilberger algorithmic technique. In this paper, we prove $q$-analogues of two of them via the $q$-WZ method. Additionally, we give $q$-analogues of two related congruence of Sun, one is confirmed and the other is conjectural.

math.NT↗

arXiv2017

Factors of sums involving $q$-binomial coefficients and powers of $q$-integers

We show that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and any non-negative integers $j$ and $r$ with $j\leqslant m$, the expression $$ \frac{1}{[n_1]}{n_1+n_{m}\brack n_1}^{-1} \sum_{k=1}^{n_1}[2k][k]^{2r}q^{jk^2-(r+1)k}\prod_{i=1}^{m} {n_i+n_{i+1}\brack n_i+k} $$ is a Laurent polynomial in $q$ with integer cofficients, where $[n]=1+q+\cdots+q^{n-1}$ and ${n\brack k}=\prod_{i=1}^k(1-q^{n-i+1})/(1-q^i)$. This gives a $q$-analogue of a divisibility result on the Catalan triangle obtained by the first author and Zeng, and also confirms a conjecture of the first author and Zeng. We further propose several related conjectures.

math.CO↗

arXiv2017

Factors of sums and alternating sums of products of $q$-binomial coefficients and powers of $q$-integers

We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and non-negative integers $j$ and $r$ with $j\leqslant m$, the following two expressions \begin{align*} &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{-1}\sum_{k=0}^{n_1} q^{j(k^2+k)-(2r+1)k}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack n_i-k},\\[5pt] &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{-1}\sum_{k=0}^{n_1}(-1)^k q^{{k\choose 2}+j(k^2+k)-2rk}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack n_i-k} \end{align*} are Laurent polynomials in $q$ with integer coefficients, where $[n]=1+q+\cdots+q^{n-1}$ and ${n\brack k}=\prod_{i=1}^k(1-q^{n-i+1})/(1-q^i)$. This gives a $q$-analogue of some divisibility results of sums and alternating sums involving binomial coefficients and powers of integers obtained by Guo and Zeng. We also confirm some related conjectures of Guo and Zeng by establishing their $q$-analogues. Several conjectural congruences for sums involving products of $q$-ballot numbers $\left({2n\brack n-k}-{2n\brack n-k-1}\right)$ are proposed in the last section of this paper.

math.NT↗

arXiv2017

A $q$-Clausen-Orr type formula and its applications

We show that certain terminating $_{6}ϕ_5$ series can be factorized into a product of two $_{3}ϕ_{2}$ series. As applications we prove a summation formula for a product of two $q$-Delannoy numbers along with some congruences for sums involving $q$-Delannoy numbers. This confirms three recent conjectures of the second author.

math.CO↗

arXiv2017

Proof of a conjecture of Kløve on permutation codes under the Chebychev distance

Let $d$ be a positive integer and $x$ a real number. Let $A_{d, x}$ be a $d\times 2d$ matrix with its entries $$ a_{i,j}=\left\{ \begin{array}{ll} x\ \ & \mbox{for} \ 1\leqslant j\leqslant d+1-i, 1\ \ & \mbox{for} \ d+2-i\leqslant j\leqslant d+i, 0\ \ & \mbox{for} \ d+1+i\leqslant j\leqslant 2d. \end{array} \right. $$ Further, let $R_d$ be a set of sequences of integers as follows: $$R_d=\{(ρ_1, ρ_2,\ldots, ρ_d)|1\leqslant ρ_i\leqslant d+i, 1\leqslant i \leqslant d,\ \mbox{and}\ ρ_r\neq ρ_s\ \mbox{for}\ r\neq s\}.$$ and define $$Ω_d(x)=\sum_{ρ\in R_d}a_{1,ρ_1}a_{2, ρ_2}\ldots a_{d,ρ_d}.$$ In order to give a better bound on the size of spheres of permutation codes under the Chebychev distance, Kløve introduced the above function and conjectured that $$Ω_d(x)=\sum_{m=0}^d{d\choose m}(m+1)^d(x-1)^{d-m}.$$ In this paper, we settle down this conjecture positively.

cs.IT↗

arXiv2017

Factors of alternating sums of powers of $q$-Narayana numbers

The $q$-Narayana numbers $N_q(n,k)$ and $q$-Catalan numbers $C_n(q)$ are respectively defined by $$ N_q(n,k)=\frac{1-q}{1-q^n}{n\brack k}{n\brack k-1}\quad\text{and}\quad C_n(q)=\frac{1-q}{1-q^{n+1}}{2n\brack n}, $$ where ${n\brack k}=\prod_{i=1}^{k}\frac{1-q^{n-i+1}}{1-q^i}$. We prove that, for any positive integers $n$ and $r$, there holds \begin{align*} \sum_{k=-n}^{n}(-1)^{k}q^{jk^2+{k\choose 2}}N_q(2n+1,n+k+1)^r \equiv 0 \pmod{C_n(q)}, \end{align*} where $0\leqslant j\leqslant 2r-1$. We also propose several related conjectures.

math.NT↗