Some q-analogues of supercongruences of Rodriguez-Villegas

We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence \sum_{k=0}^{p-1}\frac{{2k\choose k}^2}{16^k} \equiv (-1)^{\frac{p-1}{2}}\pmod{p^2} has the following two nice q-analogues with [p]=1+q+...+q^{p-1}: \sum_{k=0}^{p-1}\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{(1+\varepsilon)k} &\equiv (-1)^{\frac{p-1}{2}}q^{\frac{(p^2-1)\varepsilon}{4}}\pmod{[p]^2}, where (a;q)_0=1, (a;q)_n=(1-a)(1-aq)...(1-aq^{n-1}) for n=1,2,..., and \varepsilon=\pm1. Several related conjectures are also proposed.Comment: 14 pages, to appear in J. Number Theor

On Zudilin's q-question about Schmidt's problem

We propose an elemantary approach to Zudilin's q-question about Schmidt's problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a previous paper [Acta Arith. 127 (2007), 17--31]. The new approach is based on a q-analogue of our recent result in [J. Number Theory 132 (2012), 1731--1740] derived from q-Pfaff-Saalschutz identity.Comment: 5 page

A note on two identities arising from enumeration of convex polyominoes

Motivated by some binomial coefficients identities encountered in our approach to the enumeration of convex polyominoes, we prove some more general identities of the same type, one of which turns out to be related to a strange evaluation of ${}_3F_2$ of Gessel and Stanton.Comment: 10 pages, to appear in J. Comput. Appl. Math; minor grammatical change

A Question Selection Strategy for Early Warning Systems

Early warning systems, or early alert systems, are systems to identify students at risk of failing a course. These systems use two categories of indicators: Traditional indicators such as assignment grades and class attendance, and “soft” factors such as the student’s behavior and learning network. Naturally, in the interest of preserving user engagement, an early warning system should ask the least amount of questions possible. In this research, we seek to determine if it is possible for an academic early warning system to obtain a level of prediction accuracy from an incomplete data set like that which can be obtained from a complete data set. A set of questions is developed about the student’s study habits, study attitudes, study anxiety, time management, learning network, and class participation. The questions answers are used to identify student’s characteristics. The pilot study is based on previous sessions of the Data Structures course at the University of Houston. First, classifiers are constructed based on two different algorithms k-nearest neighbors and feed-forward-neural network algorithms. Then training datasets of assignment and exams grades are measured using the three-fold cross validation method. In the future, we plan on implementing the study by asking the students in the upcoming fall session of Data Structures these questions and perform a mutual information analysis of their responses. If there is a high level of mutual information we will perform offline experiments on the data set to explore a mutual information approach and a PCA based approach to select optimal subsets of questions to ask individual students.Honors CollegeComputer Science, Department o

Combinatorial Interpretations of the q-Faulhaber and q-Salie Coefficients

Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhaber's formula for the sums of powers and a q-analogue of Gessel-Viennot's formula involving Salie's coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel-Viennot's combinatorial interpretations.Comment: 15 page

A Generalization of the Ramanujan Polynomials and Plane Trees

Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has recently introduced a polynomial sequence Q_n:=Q_n(x,y,z,t) defined by Q_1=1, Q_{n+1}=[x+nz+(y+t)(n+y\partial_y)]Q_n. In this paper we prove Chapoton's conjecture on the duality formula: Q_n(x,y,z,t)=Q_n(x+nz+nt,y,-t,-z), and answer his question about the combinatorial interpretation of Q_n. Actually we give combinatorial interpretations of these polynomials in terms of plane trees, half-mobile trees, and forests of plane trees. Our approach also leads to a general formula that unifies several known results for enumerating trees and plane trees.Comment: 20 pages, 2 tables, 8 figures, see also http://math.univ-lyon1.fr/~gu