相关论文: A q-analogue of Lehmer's congruence

We give q-analogues of Wilson's theorem for the primes congruent 1 and 3 modulo 4 respectively. And q-analogues of two congruences due to Mordell and Chowla are also established.

In the paper, we generalize some congruences of Lehmer for general composite numbers.

Elementary proofs of Sylvester's, Wolstenholme's, Morley's and Lehmer's congruence theorems

We establish a q-analogue of Wolstenholme's harmonic series congruence.

Recently, Straub gave an interesting $q$-analogue of a binomial congruence of Ljunggren. In this note we give an inductive proof of his result.

We prove some symmetric $q$-congruences.

Andrews once gave $q$-analogues of a binomial congruence of Glaisher, and he suggested perfect $q$-analogues. In this note we give ones meeting the demand of Andrews.

We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that \[ \binom{a p}{b p} \equiv \binom{a}{b} \] modulo $p^3$ for primes $p\ge5$. This congruence subsumes and builds on earlier congruences by Babbage,…

A new $q$-analogue of Appell polynomial sequences and their generalizations are introduced and their main characterizations are proved. As consequences new $q$-analogue of Bernoulli and Euler polynomials and numbers is introduced, their…

Applying the $q$-Zeilberger algorithm, we establish a unified $q$-analogue of the (C.2) and (G.2) supercongruences of Van Hamme, which can be viewed as a refinement of several previously known results. As consequences, we obtain a…

We construct the q-analogue of Euler-Barnes' numbers and polynomials, and investigate their some properties.

We give a q-analogue of Gauss' divisibility theorem

We provide some variations on the Greene-Krammer's identity which involve q-Catalan numbers. Our method reveals a curious analogy between these new identities and some congruences modulo a prime.

In this paper we establish a $q$-analogue of a congruence of Sun concerning the products of binomial coefficients modulo the square of a prime.

We discuss $q$-analogues of the classical congruence $\binom{ap}{bp}\equiv\binom{a}{b}\pmod{p^3}$, valid for primes $p>3$, as well as its generalisations. In particular, we prove related congruences for ($q$-analogues of) integral factorial…

We establish a $q$-analogue of Sun--Zhao's congruence on harmonic sums. Based on this $q$-congruence and a $q$-series identity, we prove a congruence conjecture on sums of central $q$-binomial coefficients, which was recently proposed by…

In this paper, we study the Lehmer's type congruences for lacunary harmonic sums.

In this paper we study the genralized q-Euler numbers and polynomials. From our results, we derive some interesting congruences related tothe generalized q-Euler numbers.

We consider a family of polynomial analogs of the Ap\'ery numbers, which includes $q$-analogs of Krattenthaler--Rivoal--Zudilin and Zheng, and show that the supercongruences that Gessel and Mimura established for the Ap\'ery numbers…

We prove a Lucas-type congruence for q-Delannoy numbers.