Related papers: Some symmetric $q$-congruences
This thesis is focused on some solvable quantum mechanical models and their associated symmetries.
In this paper, we derive certain congruences for the number of $3$-core cubic bipartitions using elementary $q$-series manipulations and dissection formulas.
In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2}…
We establish $q$-analogs for four congruences involving central binomial coefficients. The $q$-identities necessary for this purpose are shown via the $q$-WZ method.
The geometric realizations of Lusztig's symmetries of symmetrizable quantum groups are given in this paper. This construction is a generalization of that in [19].
We find a $q$-analog of the following symmetrical identity involving binomial coefficients $\binom{n}{m}$ and Eulerian numbers $A_{n,m}$, due to Chung, Graham and Knuth [{\it J. Comb.}, {\bf 1} (2010), 29--38]: {equation*} \sum_{k\geq…
A novel approach to an old symmetry problem is developed. A new proof is given for the following symmetry problem, studied earlier.
We present some completely monotonic functions involving the$q$-polygamma functions, our result generalizes some known results.
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
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.
Recently, Straub gave an interesting $q$-analogue of a binomial congruence of Ljunggren. In this note we give an inductive proof of his result.
Some aspects of $Q$-conditional symmetry and of its connections with reduction and compatibility are discussed.
We demonstrate that QED exhibits a previously unobserved symmetry. Some consequences are discussed.
We introduce and study a new kind of congruent number problem on the right trapezoid.
We prove a uniformization theorem in complex algebraic geometry.
We examine the convergence of $q$-hypergeometric series when $|q|=1$.
We prove some properties of completely monotonic functions and apply them to obtain results on gamma and $q$-gamma functions.
We study quartic double solids admitting icosahedral symmetry.
Elementary proofs of Sylvester's, Wolstenholme's, Morley's and Lehmer's congruence theorems
We give a survey of results on the geometry of complex algebraic Q-acyclic surfaces, so-called 'Q-homology planes', including some recent results.