Victor J. W. Guo

Some arithmetic properties of the q -Euler numbers and q -Salié numbers

For m > n ≥ 0 and 1 ≤ d ≤ m, it is shown that the q-Euler number E2m(q) is congruent to qm-n E2n(q) mod (1 + qd) if and only if m ≡ n mod d. The q-Salie number S2n(q) is shown to be divisible by (1 + q2r+1)⌊n/2r+1⌋ for any r ≥ 0. Furthermore, similar congruences for the generalized q-Euler numbers are also obtained, and some conjectures are formulated.

Factors of alternating sums of products of binomial and q-binomial coefficients

In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\leq j\leq m-1, {n_1+n_{m}\brack n_1}^{-1}\sum_{k=-n_1}^{n_1}(-1)^kq^{jk^2+{k\choose 2}} \prod_{i=1}^m {n_i+n_{i+1}\brack n_i+k}\in \N[q], which generalizes a result of Calkin [Acta Arith. 86 (1998), 17--26]. Moreover, we show that for all positive integers n, r and j, {2n\brack n}^{-1}{2j\brack j} \sum_{k=j}^n(-1)^{n-k}q^{A}\frac{1-q^{2k+1}}{1-q^{n+k+1}} {2n\brack n-k}{k+j\brack k-j}^r\in N[q], where A=(r-1){n\choose 2}+r{j+1\choose 2}+{k\choose 2}-rjk, which solves a problem raised by Zudilin [Electron. J. Combin. 11 (2004), #R22].

Some q-analogues of supercongruences of Rodriguez-Villegas

Abstract We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence ∑ k = 0 p − 1 ( 2 k k ) 2 16 k ≡ ( − 1 p ) ( mod p 2 ) , where ( ⋅ p ) is the Legendre symbol, has the following two nice q-analogues: ∑ k = 0 p − 1 ( q ; q 2 ) k 2 ( q 2 ; q 2 ) k 2 q ( 1 + e ) k ≡ ( − 1 p ) q ( p 2 − 1 ) e 4 ( mod ( 1 + q + ⋯ + q p − 1 ) 2 ) , where ( a ; q ) n = ( 1 − a ) ( 1 − a q ) ⋯ ( 1 − a q n − 1 ) and e = ± 1 . Several related conjectures are also proposed.

Some divisibility properties of binomial and q-binomial coefficients

Abstract We first prove that if a has a prime factor not dividing b then there are infinitely many positive integers n such that ( a n + b n a n ) is not divisible by b n + 1 . This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that ( 12 n 3 n ) and ( 12 n 4 n ) are divisible by 6 n − 1 , and that ( 330 n 88 n ) is divisible by 66 n − 1 , for all positive integers n . As we show, the latter results are in fact consequences of divisibility and positivity results for quotients of q -binomial coefficients by q -integers, generalising the positivity of q -Catalan numbers. We also put forward several related conjectures.

Factors of binomial sums from the Catalan triangle

Abstract 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 n 1 ) − 1 ∑ k = 1 n 1 k 2 r + 1 ∏ i = 1 m ( n i + n i + 1 n i + k ) is either an integer or a half-integer. Moreover, several related conjectures are proposed.

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 p@?(n) and for the partition function pod(n) counting the partitions of n with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.

Proof of some conjectures of Z.-W. Sun on congruences for Apéry polynomials

Abstract The Apery polynomials are defined by A n ( x ) = ∑ k = 0 n ( n k ) 2 ( n + k k ) 2 x k for all nonnegative integers n. We confirm several conjectures of Z.-W. Sun on the congruences for the sum ∑ k = 0 n − 1 ( − 1 ) k ( 2 k + 1 ) A k ( x ) with x ∈ Z .

Short proofs of summation and transformation formulas for basic hypergeometric series

We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobis triple product identity and Watsons quintuple product identity are also proved.

Proof of Sun's conjectures on integer-valued polynomials

Abstract Recently, Z.-W. Sun introduced two kinds of polynomials related to the Delannoy numbers, and proved some supercongruences on sums involving those polynomials. We deduce new summation formulas for squares of those polynomials and use them to prove that certain rational sums involving even powers of those polynomials are integers whenever they are evaluated at integers. This confirms two conjectures of Z.-W. Sun. We also conjecture that many of these results have neat q-analogues.

Elementary proofs of some q-identities of Jackson and Andrews-Jain

We present elementary proofs of three q-identities of Jackson. They are Jacksons terminating q-analogue of Dixons sum, Jacksons q-analogue of Clausens formula, and a generalization of both of them. We also give an elementary proof of Jains q-analogue of terminating Watsons summation formula, which is actually equivalent to Andrewss q-analogue of Watsons formula.