Related papers: Some symmetric $q$-congruences
Recently the new q-Euler numbers are defined. In this paper we derive the the Kummer type congruence related to q-Euler numbers and we introduce some interesting formulae related to these q-Euler numbers.
We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable.…
In this article we establish some properties regarding the solutions of a linear congruence, bases of solutions of a linear congruence, and the finding of other solutions starting from these bases.
We obtain a small improvement of Gallagher's larger sieve and we extend it to higher dimensions. We also obtain two interesting upper bounds for the number of solutions to polynomial congruences.
We give q-analogues of the following congruences by Z.-W. Sun: \sum_{k=1}^{p-1}\frac{D_k}{k} \equiv -\frac{2^{p-1}-1}{p} \pmod p,\\ \sum_{k=1}^{p-1}\frac{H_k}{k 2^k}\equiv 0 \pmod{p},\quad p\geqslant 5, where p is a prime,…
We prove several Stern's type congruences for generalized bernoulli numbers.
In the paper, we generalize some congruences of Lehmer for general composite numbers.
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 give a study of the symmetrized divergences $U_s(p,q)=K_s(p||q)+K_s(q||p)$ and $V_s(p,q)=K_s(p||q)K_s(q||p)$, where $K_s$ is the relative divergence of type $s, s\in\mathbb R$. Some basic properties as symmetry,…
We prove some restriction theorems for flat homogeneous surfaces of codimension greater than one.
We prove an inequality for polynomials applied in a symmetric way to non-commuting operators.
We prove some Rodriguez-Villegas type congruences for truncated q-hypergeometric functions.
We prove some identities for the squares of generalized Tribonacci numbers. Various summation identities involving these numbers are derived.
We present some completely monotonic functions involving the $q$-gamma function that are inspired by their analogues involving the gamma function.
The main object of this paper is to obtain several symmetric properties of the q-Zeta type functions. As applications of these properties, we give some new interesting identities for the modified q-Genocchi polynomials. Finally, our…
We prove some general theorems for preserving Dependent Choice when taking symmetric extensions, some of which are unwritten folklore results. We apply these to various constructions to obtain various simple consistency proofs.
We prove the consistency of the existence of a $Q$-set whose square is not a $\Delta$-set and that if there is a $\Delta$-set, then there exists a $\Delta$-set whose all finite powers are $\Delta$-sets.
We study embeddings of symmetric groups to the space Cremona group.
We introduce the $\alpha,\beta$-symmetric difference derivative and the $\alpha,\beta$-symmetric N\"orlund sum. The associated symmetric quantum calculus is developed, which can be seen as a generalization of the forward and backward…
We exploit symmetries to give short proofs for two prominent formula families of QBF proof complexity. On the one hand, we employ symmetry breakers. On the other hand, we enrich the (relatively weak) QBF resolution calculus Q-Res with the…