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Victor J. W. Guo

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

At least 73 records · Page 4Linked to original sources
arXiv2013

Contiguous relations and summation and transformation formulas for basic hypergeometric series

By using contiguous relations for basic hypergeometric series, we give simple proofs of Bailey's $_4ϕ_3$ summation, Carlitz's $_5ϕ_4$ summation, Sears' $_3ϕ_2$ to $_5ϕ_4$ transformation, Sears' ${}_4ϕ_3$ transformations, Chen's bibasic summation, Gasper's split poised $_{10}ϕ_9$ transformation, Chu's bibasic symmetric transformation. Along the same line, finite forms of Sylvester's identity, Jacobi's triple product identity, and Kang's identity are also obtained.

math.CO↗
arXiv2013

Two truncated identities of Gauss

Two new expansions for partial sums of Gauss' triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$ counting the partitions of $n$ with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.

math.CO↗
arXiv2012

Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers

For all nonnegative integers n, the Franel numbers are defined as $$ f_n=\sum_{k=0}^n {n\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0 \pmod{2n^2}, \sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\equiv 2p^2 (2^p-1)^2 \pmod{p^5}, where n is a positive integer and p>3 is a prime.

math.NT↗
arXiv2012

On Zudilin's q-question about Schmidt's problem

We propose an elemantary approach to Zudilin's q-question about Schmidt's problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a previous paper [Acta Arith. 127 (2007), 17--31]. The new approach is based on a q-analogue of our recent result in [J. Number Theory 132 (2012), 1731--1740] derived from q-Pfaff-Saalschutz identity.

math.CO↗
arXiv2012

New congruences for sums involving Apery numbers or central Delannoy numbers

The Apéry numbers $A_n$ and central Delannoy numbers $D_n$ are defined by $$A_n=\sum_{k=0}^{n}{n+k\choose 2k}^2{2k\choose k}^2, \quad D_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}. $$ Motivated by some recent work of Z.-W. Sun, we prove the following congruences: \sum_{k=0}^{n-1}(2k+1)^{2r+1}A_k &\equiv \sum_{k=0}^{n-1}\varepsilon^k (2k+1)^{2r+1}D_k \equiv 0\pmod n, where $n\geqslant 1$, $r\geqslant 0$, and $\varepsilon=\pm1$. For $r=1$, we further show that \sum_{k=0}^{n-1}(2k+1)^{3}A_k &\equiv 0\pmod{n^3}, \quad \sum_{k=0}^{p-1}(2k+1)^{3}A_k &\equiv p^3 \pmod{2p^6}, where $p>3$ is a prime. The following congruence \sum_{k=0}^{n-1} {n+k\choose k}^2{n-1\choose k}^2 \equiv 0 \pmod{n} plays an important role in our proof.

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arXiv2011

A q-analogue of some binomial coefficient identities of Y. Sun

We give a $q$-analogue of some binomial coefficient identities of Y. Sun [Electron. J. Combin. 17 (2010), #N20] as follows: {align*} \sum_{k=0}^{\lfloor n/2\rfloor}{m+k\brack k}_{q^2}{m+1\brack n-2k}_{q} q^{n-2k\choose 2} &={m+n\brack n}_{q}, \sum_{k=0}^{\lfloor n/4\rfloor}{m+k\brack k}_{q^4}{m+1\brack n-4k}_{q} q^{n-4k\choose 2} &=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k{m+k\brack k}_{q^2}{m+n-2k\brack n-2k}_{q}, {align*} where ${n\brack k}_q$ stands for the $q$-binomial coefficient. We provide two proofs, one of which is combinatorial via partitions.

math.CO↗
arXiv2011

Factors of sums and alternating sums involving binomial coefficients and powers of integers

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}ε^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$ \sum_{k=0}^{n}ε^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s} \equiv 0 \mod{2n\choose n}, $$ where $ε=\pm 1$.

math.NT↗
arXiv2010

Bijective proofs of Gould-Mohanty's and Raney-Mohanty's identities

Using the model of words, we give bijective proofs of Gould-Mohanty's and Raney-Mohanty's identities, which are respectively multivariable generalizations of Gould's identity $$\sum_{k=0}^{n}{x-kz\choose k}{y+kz\choose n-k}= \sum_{k=0}^{n}{x+ε-kz\choose k}{y-ε+kz\choose n-k} $$ and Rothe's identity $$ \sum_{k=0}^{n}\frac{x}{x-kz}{x-kz\choose k}{y+kz\choose n-k}= {x+y\choose n}. $$

math.CO↗
arXiv2010

Bijective Proofs of Gould's and Rothe's Identities

We first give a bijective proof of Gould's identity in the model of binary words. Then we deduce Rothe's identity from Gould's identity again by a bijection, which also leads to a double-sum extension of the $q$-Chu-Vandermonde formula.

math.CO↗
arXiv2010

Simple proofs of Jensen's, Chu's, Mohanty-Handa's, and Graham-Knuth-Patashnik's identities

Motivated by the recent work of Chu [Electron. J. Combin. 17 (2010), #N24], we give simple proofs of Jensen's identity $$ \sum_{k=0}^{n}{x+kz\choose k}{y-kz\choose n-k} =\sum_{k=0}^{n}{x+y-k\choose n-k}z^k, $$ and Chu's and Mohanty-Handa's generalizations of Jensen's identity. We also give a quite simple proof of an equivalent form of Graham-Knuth-Patashnik's identity $$ \sum_{k\geq 0}{m+r\choose m-n-k}{n+k\choose n}x^{m-n-k}y^k =\sum_{k\geq 0}{-r\choose m-n-k}{n+k\choose n}(-x)^{m-n-k}(x+y)^k, $$ which was rediscovered, respectively, by Sun in 2003 and Munarini in 2005. Finally we give a multinomial coefficient generalization of this identity and raise two open problems.

math.CO↗
arXiv2010

Some congruences involving central q-binomial coefficients

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$ \sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5}) q^{-\lfloor n^4/5\rfloor} \pmod{Φ_n(q)}, $$ where $\big(\frac{n}{p}\big)$ is the Legendre symbol and $Φ_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that $$ \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, $$ for $a,m\geq 1$, the first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence modulo powers of 3. Several related conjectures are proposed.

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arXiv2009

Factors of binomial sums from the Catalan triangle

By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers $n_1, ..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, the expression $$n_1^{-1}{n_1+n_{m}\choose n_1}^{-1} \sum_{k=1}^{n_1}k^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}\choose n_i+k}$$ is either an integer or a half-integer. Moreover, several related conjectures are proposed.

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arXiv2009

Pairs of lattice paths and positive trigonometric sums

Ismail et al. (Constr. Approx. {\bf 15} (1999) 69--81) proved the positivity of some trigonometric polynomials with single binomial coefficients. In this paper, we prove some similar results by replacing the binomial coefficients with products of two binomial coefficients.

math.CO↗
arXiv2009

On the least common multiple of $q$-binomial coefficients

In this paper, we prove the following identity $$ \lcm({n\brack 0}_q,{n\brack 1}_q,...,{n\brack n}_q) =\frac{\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q}, $$ where ${n\brack k}_q$ denotes the $q$-binomial coefficient and $[n]_q=\frac{1-q^n}{1-q}$. This result is a $q$-analogue of an identity of Farhi [Amer. Math. Monthly, November (2009)].

math.NT↗