Related papers: q-Analogue of Gauss' Divisibility Theorem

In this paper, the divisibility property of the type 2 $(p, q)$-analogue of the $r$-Whitney numbers of the second kind is established. More precisely, a congruence relation modulo $pq$ for this $(p,q)$-analogue is derived.

We establish the q-analogue of a classical congruence of Lehmer. Also, the q-analogues of two congruences of Morley and Granville are given.

Following an idea due to J. Bernoulli, we explore the q-analogue of the sums of powers of consecutive integers.

We provide a $q$-analogue of Euler's formula for $\zeta(2k)$ for $k\in\mathbb{Z}^+$. Our main results are stated in Theorems 3.1 and 3.2 below. The result generalizes a recent result of Z.W. Sun who obtained $q$-analogues of…

We give a $q$-analogue of $\zeta(6)=\pi^6/945$. Our main results are stated in Theorems 2.1 and 2.2 below.

If we consider a q-analogue of linear differential equation, Galoois group of the q-analogue difference equation is still a linear algebraic group. Namely, by a quantization of linear differential equation, Galois group is not quantized. We…

In this paper we shall evaluate two alternating sums of binomial coefficients by a combinatorial argument. Moreover, by combining the same combinatorial idea with partition theoretic techniques, we provide $q$-analogues involving the…

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

Recently, the concept of a D-analogue was introduced by the author. This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series. we consider the D-analogues of the q-binomial coefficients, and a…

In this paper, we establish a q-analog of partial fraction decomposition formula. By using formula, we develop new closed form representations of sums of q-harmonic numbers and reciprocal q-binomial coefficients. Moreover, we give explicit…

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.

In this paper, we consider the q-extensions of Boole polynomials. From those polynomials, we derive some new and interesting properties and identities related to special polynomials.

I present a $q$-analog of the discrete Painlev\'e I equation, and a special realization of it in terms of $q$-orthogonal polynomials.

We derive a $q$-analogue of the matrix sixth Painlev\'e system via a connection-preserving deformation of a certain Fuchsian linear $q$-difference system. In specifying the linear $q$-difference system, we utilize the correspondence between…

We study some q-analogues of the Racah polynomials and some of their applications in the theory of representation of quantum algebras.

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…

A q-analogue of de Finetti's theorem is obtained in terms of a boundary problem for the q-Pascal graph. For q a power of prime this leads to a characterisation of random spaces over the Galois field F_q that are invariant under the natural…

We introduce four q-analogs of the double Laplace transform and prove some of their main properties. Next we show how they can be used to solve some q-functional equations and partial q-differential equations.

From two q-summation formulas we deduce certain series expansion formulas involving the q-gamma function. With these formulas we can give q-analogues of series expansions for certain constants.

We propose in this paper a Galois theory of $q$-difference equations where q is a root of unity. This theory is the q difference analogue of the Galois theory of iterative differential equations, that is differential equations over fields…