Related papers: Some symmetric $q$-congruences

Inspired by the recent work on $q$-congruences and the quadratic summation formula of Rahman, we provide some new $q$-supercongruences. By taking $q\to 1$ in one of our results, we obtain a new Ramanujan-type supercongruence, which has the…

We prove that several results of lineability/spaceability in the framework of sequence spaces are valid in a stricter sense.

We improve on the spectral large sieve inequality for symmetric-squares. We also prove a lower bound showing that the most optimistic upper bound is not true for this family.

Our main result is the construction of symmetric Hadamard matrices of order q(1 + q) where q is a prime power congruent to 3 mod 8.

We describe a congruence property of solvable polynomials over Q, based on the irreducibility of cyclotomic polynomials over number fields that meet certain conditions.

A symplectic realization and some symmetries of a Rikitake type system are presented.

We prove orthogonality relations for some analogs of trigonometric functions on a $q$-quadratic grid and introduce the corresponding $q$-Fourier series. We also discuss several other properties of this basic trigonometric system and the…

In this paper, we describe a congruence property of solvable polynomials with coefficients in the Gaussian field Q(i).

We introduce extremely symmetric primes and provide some elementary properties of these.

We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercongruences.

We prove some special cases of Bergeron's inequality involving two Gaussian polynomials (or $q$-binomials).

The cyclic sieving phenomenon provides a link between a polynomial analogue of Gauss congruence known as $q$-Gauss congruence, and a combinatorial analogue of Gauss congruence based on sequences of cyclic group actions. We strengthen this…

We prove some 3-adic congruences for binomial sums, which were conjectured by Sun.

We construct a q-analogue of truncated version of symmetric multiple zeta values which satisfies the double shuffle relation. Using it, we define a q-analogue of symmetric multiple zeta values and see that it satisfies many of the same…

In this article we determine several theorems and methods for solving linear congruences and systems of linear congruences, and we find the number of distinct solutions. Many examples of solving congruences are given.

Using Watson's terminating $_8\phi_7$ transformation formula, we prove a family of $q$-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019),…

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Ap\'{e}ry numbers, by means of a general…

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…

Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also…

In this paper, we establish a $q$-integral formula by using the orthogonality relation, and also provide a new proof of the $q$-orthogonality relation for the continuous $q$-ultraspherical polynomials. A new $q$-beta integral with five…